Upgrade or Replace Late 2013 iMac?

Have you tried Erasing the drive and Installing macOS from scratch? Lot cheaper and doesn't take too much of your valuable time. Hard to select the best option for you. We don't know how much time each of your options require, and what an hour of your time is worth. We also don't know what the component costs will be for each option.

Ways to Help Make a Slow Mac Faster

1. 17 Reasons Why Your Mac Runs Slower Than it Should
2. Slow Mac Performance? This Article Solves It!
3. Fix slow start-ups in OS X | MacFixIt - CNET Reviews
4. How to fix slow shutdown and startup times. | MacTip.net
5. 6 Easy Tips to Speed Up OS X Yosemite on Your Mac
6. OS X El Capitan- If your Mac runs slowly
7. Visit The XLab FAQs and read the FAQ on Performance
8. Tips to Fix Issues that affect Mac Performance
9. Avoid using any third-party cleanup software. Usually, this software does more bad than good. You don't need it. All computers become slower over time even under normal use. Experienced users erase the hard drive and do a clean install from scratch from time to time; or whenever installing a major OS upgrade. Doing so means you must maintain regular and multiple backups.
10. If you have enabled iCloud Disk and are storing your Documents and Data in iCloud, then consider turning that off and signing out of iCloud. It can slow down the computer considerably. Please see the following from the user, fotomac: "The solution was to SIGN OUT of iCloud and my problem STOPPED! NO MORE SPINNING BEACHBALL! My computer's speed increased to what it should be and all my Apps now work!"
11. Add more RAM or cut back on the number of concurrently running applications and utilities. Remove unnecessary anti-malware software and any software that promises to clean your Mac. Check for runaway processes: Runaway applications can shorten battery runtime, affect performance, and increase heat and fan activity. Also, visit The XLab FAQs and read the FAQ on dealing with The Spinning Beach Ball of Death.
12. The Ultimate Fix: Backup everything, erase the drive, reinstall OS X, and restore your data from the backup. Reinstall third-party software from original media/scratch.

Replacing RAM is a major operation in that model iMac. However,

you can get some performance improvement by migrating your system

to an external SSD boot drive.

The first option there would be either using Thunderbolt or USB 3 interface.

Thunderbolt will maximizes the performance of any SSD but will be a pricier

solution. USB3 based drive will not be quite as fast, but will still be much faster

than the slow internal HDD that is inside your iMac. Either approach should

give a a noticeable bump in performance with the bonus of not cracking

open your iMac. Either approach you would decide on, I suggest a buying

enclosure and stuffing it with a desired SSD rather that the "integrated" off

the shelve solution. If at some time in the future if you decide to crack

open your iMac, you could simply take the drive out of the enclosure and

stuff it in the iMac.

Check out OWC for their range of enclosures and drives:

https://eshop.macsales.com/shop/storage

If it were me, this would be step #1 I would take as it still has the path

to get all internal in the future if desired.

If it was fast and now it is slow it may be adware or some third party app that's slowing you down like an anti-virus or so-called "cleaning" app. You can run Etrecheck  It is a diagnostic tool that's very useful to us in finding problems. Also it will give us further specs on your Mac. After it runs post the log file here. It will contain no personal information.

To post the log file copy & paste it using the Page icon in your reply window:

A Troubleshooting Procedure that may Fix Problems with macOS El Capitan or Later

You should try each, one at a time, then test to see if the problem is fixed before going on to the next.

    Be sure to backup your files before proceeding if possible.

Shutdown the computer, wait 30 seconds, restart the computer.

Disconnect all third-party peripherals.

Resetting your Mac's PRAM and NVRAM

Reset the System Management Controller (SMC)

Reset your Startup Disk and Sound preferences.

Start the computer in Safe Mode. Test in safe mode to see if the problem persists, then restart normally.

Use Apple Hardware Test to see if there is any hardware malfunction.

Repair the disk by booting from the Recovery HD. Immediately after the chime hold down the Command and R keys until the Utility Menu appears. Choose Disk Utility and click on the Continue button. Select the indented (usually, Macintosh HD) volume entry from the side list.  Click on the First Aid button in the toolbar. Wait for the Done button to appear. Quit Disk Utility and return to the Utility Menu. Restart the computer from the Apple Menu.

Repair permissions on the Home folder: Resolve issues caused by changing the permissions of items in your home folder.

Create a New User Account Open Users & Groups preferences. Click on the lock icon and enter your Admin password when prompted. On the left under Current User click on the Add [+] button under Login Options. Setup a new Admin user account. Upon completion log out of your current account then log into the new account. If your problems cease, then consider switching to the new account and transferring your files to it - Transferring files from one User Account to another.

Download and install the OS X El Capitan 10.11.6 Combo Update or 10.12.6 Combo Update or Download macOS High Sierra 10.13.5 Combo Update or Get MacOS Mojave now from the Mac App Store as needed.

Reinstall OS X by booting from the Recovery HD using the Command and R keys. When the Utility Menu appears select Reinstall OS X then click on the Continue button.

Erase and Install OS X Restart the computer. Immediately after the chime hold down the Command and R keys until the Apple logo appears. When the Utility Menu appears:

     1. Select Disk Utility from the Utility Menu and click on the Continue button.

     2. When Disk Utility loads select the drive (out-dented entry) from the Device list.

     3. Click on the Erase icon in Disk Utility's toolbar. A panel will drop down.

     4. Set the Format type to APFS (for SSDs only) or Mac OS Extended (Journaled.)

     5. Click on the Apply button, then wait for the Done button to activate and click on it.

     6. Quit Disk Utility and return to the Utility Menu.

     7. Select Reinstall OS X and click on the Continue button.

Continued from above:

14. If none of the above helps then see How to Downgrade macOS High Sierra and macOS Reversion- 

      How to Downgrade from High Sierra.

15. If you get here without success then it's time to make an appointment at the Apple Genius Bar for 

      service. If you need to find an Apple Store - Find a Store - Apple.

Well, the procedure is to determine the total costs of choosing each option. Then you must allocate those costs over the expected lifetime of the machine for each selected option. Choose an appropriate discount rate and discount the projected costs for each option. Select the option with the highest present value.

If you want to minimize the costs of keeping the computer, then you need to perform an Erase and Install which should return the existing machine to its level of performance when new.

Personally, I would only return your files from your backup. Reinstall the third-party software from original media being sure to use only the latest versions so they are compatible with the current versions of macOS. Alternatively, take the time to go thru the procedures I posted to see if they improved performance without having to erase everything. But my opinion is an Erase and Install is the best thing to restore a computer to its "as new" level of performance. IMHO, it should be done at least once a year of prior to installing a new upgrade (not update) of the OS.

The old saw, "if it's not broke, don't fix it" applies here. As long as you have sufficient free space on your hard drive, I wouldn't touch a thing. If you need the space and do try to slim it further, it would depend on which folders inside your 2 users are the large ones. If it's Documents, Movies, Music, Pictures then you can pare it down by copying larger, lesser used files off to other external media. I wouldn't mess with Library.