Time Capsule not connecting after move...

At the old house, it connected via wifi as the router was in the other room

The older versions of the Time Capsule....like the A1409 model that you have...... and older 5.5 and 5.6 versions of AirPort Utility did provide an option to have the Time Capsule connect to the network using only a wireless connection. This setting was called "Join a wireless network".

Unfortunately, Apple removed this option when they introduced the 6.3.x versions of AirPort Utility so you will not be able to set up the Time Capsule with your new WiFi router the way that you did in the past.

Instead, the Time Capsule will need to connect to your Verizon router using a permanent, wired Ethernet cable connection. In other words, an Ethernet cable will connect from one of the four LAN <--> Ethernet ports on the Verizon router to the WAN "O" port on the Time Capsule.

In addition, the Time Capsule will need to be reset back to its factory default settings to erase the old setup.....(that you used with your previous router)......and then set up again with new settings so that it will connect correctly to your new router.

The information that Tesserax has already provided will basically walk you through the new setup. If you need additional help or need step by step instructions, please post back to let us know.

Finally, just a note here that the A1409 version of the Time Capsule is at least 6 years old and as much as 8+ depending on when you purchased the product and placed it into service. The average useful life of this model is about 5 years, so you may need to begin to think about a replacement at some point in the future.

Any time you change ISPs or networking hardware, or both, it is important to perform a complete power recycle of your networking hardware. Please see the following article for more details:

• AirPort - Complete Power Recycle - Apple Community

To avoid confusion and multiple other discussions, my suggestion would be to go ahead and get the Time Capsule set up and running with a new configuration with your new router.

Then.....we can tackle your other questions.

Nice work! You are all set if you want the Time Capsule to also provide a second wireless network. If you don't need for the Time Capsule to provide a wireless network function, then you can simply ignore that network, or you do have the option to turn off the WiFi on the Time Capsule and use it as a wired hard drive with Verizon router.

As far as your other questions......

So, even though I didn't update anything on the computer between houses, just the fact that it's a different wifi connection now means that the change to AirPort Utility will keep it from connecting?

I think you already know the answer to this one.

I suppose it doesn't matter anyways, as I've already got it connected via a cable since it is close enough to the new router to be able to do that.

Correct. Even if you could configure the Time Capsule to connect using only wireless to the new router......(you can't)......this is a poor way to connect the Time Capsule to the network since it will result in a much slower connection than a wired connection.

Is that factory reset the one where my data stays intact and will still be there once it's newly set up?

Any kind of reset......Factory Default Reset, Hard Reset or Soft Reset will not affect any of the data stored on the Time Capsule hard drive. The only way that you can lose the data on the Time Capsule hard drive is perform a completely separate and deliberate "Erase Disk" procedure.

This TC has only been in use for about 3 years, but was probably purchased about 7 years ago.

Then, you should be able to get another 3-4 years use out of the Time Capsule before you need to begin to think about a replacement product.

I read that they've now discontinued the whole product line, so what are people looking at replacing it with?

There are no other products provided by other manufacturers that combine a WiFi router with a built in hard drive. The closest that you can come to a replacement in that sense is another router like those provided by Synology or Asus that will allow you to connect a USB hard drive to the router and back up over the network that way.

Another option.....if you already have a router......simply add a Network Attached Storage (NAS) drive to the router using an Ethernet cable connection. The NAS products from Synology all support Time Machine backups.

The 7.8.1 update included some important security fixes, so you would want to perform that update.

Is that factory reset the one where my data stays intact and will still be there once it's newly set up? And I can just follow the instructions at Tesserax's link even though they're not specifically for my TC?

Yes. None of the AirPort reset types will affect the data already on the TC's internal (or external) drives.

Are there guides somewhere for how to set it all up if we want to use it as a shared drive for media?

Yes, but you will need to use your friend Google to find articles about this, since Apple provides nothing in the way of help on this topic. An example:

https://www.macworld.com/article/3263691/how-to-make-time-capsule-and-networked-drives-appear-for-your-mac.html

If we want to have our iPhones and the MacBook back up to it as well, do they have to be using it for their WiFi connection too?

The MacBook and other Macs can back up over the Verizon WiFi network.

iPhones and iPads cannot back up to a Time Capsule. They back up to either iTunes on your Mac or to iCloud.

https://support.apple.com/en-us/HT203977