iMac running slow; hard drive about half full; can't sign in to Mail accounts; Safari dragging

There may be several things going on. Hard to tell without being there in person.

If you are able to, download and run Etrecheck and paste the output here using the Additional text button below. It looks like a piece of paper with a corner folded. Etrecheck output provides information about your hardware and software installations, but no identifiable information (you can verify this before pasting in the output). Etrecheck may allow people reading this to diagnose your problem(s) and devise a fix.

I know some people using Monterey on a mechanical HDD with APFS, most find it intolerable, but some are ok with it. Apple considers a 7-year old computer "obsolete," it would put priority on the newer units with SSDs and the demands of a modern file system (APFS) and the latest operating system (Monterey) do result in older mechanical drives running very slowly, relatively speaking. Does yours have an SSD or an HDD? You can tell with About This Mac, Etrecheck will also show the type of drive. By the way, a failing internal drive (not uncommon after 7 years) will also exhibit the symptoms you have experienced, Etrecheck also provides information about the health of the drive. You can also boot into Recovery and run Disk Utility to check your drive's file system. To check the hardware health, download and run DriveDX.

My wife's iMac was a 2015 model and with a mechanical HDD, was extremely slow. I replaced her internal drive with a 1 TB SSD, the total cost was about $150, including the drive. Replacing the drive yourself is possible but very intricate and challenging so I had a professional (Apple Authorized) do it. It's like a new computer now, maybe 10x or 20x faster than it was before.

https://bombich.com/blog/2019/09/12/analysis-apfs-enumeration-performance-on-rotational-hard-drives

For your email, are the AOL accounts accessible through the AOL web interface? If yes, then your email is all there, but something in the Mac configuration is not right. Is the email coming through an IMAP account? If it is an IMAP account, you can try deleting the email accounts and recreating them so everything in the email is starting fresh. For an IMAP account, it should simply redownload all the email on the server to get you up to date, which may take a while.

Two things:

1) You deleted files and now Mail and other stuff doesn’t work. Try booting into Recovery and reinstall the MacOS. It probably won’t help, however, in which case I would consider erasing the internal drive, reinstalling a fresh MacOS, then migrate just user accounts and files (no applications, no settings, no other items) from your most recent backup. You need an RELIABLE backup for this. Alternatively, can you describe exactly what you deleted? If it included folders or files from your Library, then the associated applications may stop working. Can you move those files and folders back to their original locations?

2) For slowness, if you have a mechanical drive (HDD) or fusion drive, those run very slowly under Monterey with APFS drive configuration (APFS is mandatory for Monterey). The only solution is to use an external SSD as your boot drive or replace the internal HDD with an SSD.

The suggestion to run Etrecheck is in case something else that was installed is causing some of the problem. My wife's 2015 iMac with a mechanical drive was still usable, just very slow. We did not have those issues with passwords taking too long to enter or be recognized. But our slowness went away with the new SSD.

Apple Stores might not be willing to work on an "obsolete" computer due to policy and to no longer stocking the needed parts. You should ask before going. The independent Apple Authorized Service Providers do work on older computers and can help you locate the appropriate new SSD. AASP also seem to sometimes be lower cost than Apple Stores. Apple's online support section of their web site often lists AASP providers and sometimes you can even make an appointment with one through the Apple system.

If it were me, I would run Etrecheck and post the results here and also check the disk health via Disk Utility and DriveDX, before scheduling a disk replacement. Just in case the issue is actually something else. For us (and apparently many who have posted here), the new SSD was the answer. The iMac boot time went from 3-5 minutes to about 20 seconds with the new SSD.