A little background:

My house was built in 1890. It's an old corner store that was converted into a duplex. We've been in the house about a year and a half, and of course with age there's a lot to learn. We haven't had any real major problems but I'm also not the most handy man there is. I am excited and open to learning what I can do around the house vs what I need to just call a professional for... and so far most of the issues we've encountered have been pretty straightforward.

The situation:

What's going on right now is in the shower. It started as a latent drip which has gradually over the last week or two gotten much more steady, even fast. I did a search online and have watched a bunch of videos on how to do this "relatively easy home repair" and outside of the hardware I'm working with being very different from those videos it looks simple enough to be sure... but still I have some questions and I'm hoping more experienced folks here may be able to help with that.

From what I can tell, the faucet is an Eljer brand. That's engraved on the bottom piece that has the little switch to close off the tub (that piece has never worked since we got the house, for what it's worth - we just use a separate drain stopper if the kids want a bath). I can't find the exact model of what I'm working with anywhere online. It's a single turn style handle. It doesn't pull out or anything you just turn it and a little turn gets you cold and fully turned to the right is full blast hot water.

This handle doesn't even have an access screw to remove it as the tons of videos I've found online always tell me to start with... instead what I've found by just messing with it is that you actually rotate the fixture between the use handle and the wall plate around and round and it unscrews from the base that way. Inside is a piece that attaches to the valve and allows to turn the valve from the actual handle to get the hot/cold water going.

I've been able to remove it (and I've since replaced it) but I'm not sure if the valve inside is universal with other types - every video I've seen so far indicates that this sort of leak is rooted in needing to replace that valve though. This one looks smaller. I can't find the pin... and I haven't gotten deep enough into dismantling it to actually shut off the house water and all that, so I hesitate to push too far into unknown territory... which is why I am here.

Can anyone point me in a productive direction? Are you familiar with what I've described and can you direct me to any resources, replacement parts, etc. that I am probably going to need to do this repair?

Thanks for your time in reading this long post and any help you can provide. I look forward to it!