Coming back to give a few updated bits of info on this. Sorry if this a bit rambley. (Also, thank you to the people who posted after my last message; don't think I got notified of your replies.)
For starters, I found a copy of the ad from ten years ago when I bought this car. The ad said the AC compressor was three-years-old (at that time) but the AC wasn't blowing cold. The ad recommended that the system be converted to R-134a and recharged (suggesting the only problem was a small leak). So I'm thinking the compressor locked up from sitting idle for the last decade (since it's never even had a belt on it since I got the car).
That bit of backstory aside, I had to call quite a few shops before I found one that would work on the car, but I just finished up there today. The mechanic was very nice and let me watch him work so I could learn more about tackling AC repairs myself in the future. I told him what I knew, brought in my extra compressor and some R12 cans (since he had none on hand), and let him have at it.
When he started, there was no refrigerant left in the system but he was able to pull a good vacuum. He pointed out that oil had leaked from the back of the old compressor and based on how insignificant the leak seemed to be, it was hopefully just from bad seals on the two line ports there. My extra compressor had a semi-stiff pulley, so he took the good pulley off the frozen compressor, swapped those, put new seals on the back, and put the replacement compressor in. While the compressor was removed, he looked inside the lines for debris and saw none, so he said he felt no need to replace any other parts. He added some oil, which he said had some leak detection dye in it, and then started adding the R12. This is where we ran into a slight problem. I brought four 14oz cans, but two of them leaked while attempting to add. One of them was really bad and I'm willing to bet none of it actually made it into the system. After he added what I had, he said the high pressure line had the pressure he wanted, but the low pressure line was a little lower than he'd like to see. He said if it were his car, he'd probably buy a bit more R12 to add. The temperature is operably cold, but certainly nothing to write home about. He said run it and if the performance drops, bring it back in because there is a leak somewhere other than the seals he replaced (which he'll hopefully be able to find due to the dye in the oil).
I'm a little leery that he didn't replace any other parts based on what people have said here, but I guess at this point I'm just playing a waiting game since the job is already done and the system seems to be working. I'm not sure if I want to add a bit more R12 like he suggested right away, or wait and see if the system leaks to minimize wasted refrigerant. I think I'm going to buy a thermometer to monitor the temperature at the vents instead of just trying to go off how it feels to my hand (tried to hold a disconnected wall thermostat in front of a vent and got 51°, but I'm not sure I trust that number). Out of curiosity, does anyone here know what the capacity of the system in my car is?