Speedometer and A/C problems, rough idle and stall problems

-1990 Lumina 3.1L Euro
I first noticed the problem when I was a stoplight. Car idled erratic and a few minutes later it died. When trying to start the car, I would have to step on the accelerator to start it and continue to gas it to keep rpm high. This problem is sporadic as it would work for awhile without having to gas the car but other times, I would have gas the car to start it and it would die if I let go of the accelerator. Also, sometimes when the car is started, it would idle high for a few minutes and then drop back to it's normal idle. When driving the car and coming to a stop, I would have to switch to neutral so that I can step on the accelerator to keep the car alive. Otherwise, it would die when coming to a stop when in drive. To complicate things, the 2nd day I drove the car, I notice the speedometer/odomter and a/c stopped working. I parked the car and left it alone. When I went back to it hours later, a/c and speedo/odometer still weren't working but I drove it home and eventually both started working again. Similar situation happened the 3rd day. A/C and speedometer didn't work at first but eventually both work at the same time. So it looks like the A/C and speedometer/odometer are related. The last complication came about 2 days ago. The battery indicator (volts) in the car lit up. I continued to drive the car and noticed everything getting dimmer. Just as I arrived home, the car died. When I tried to start the car, it was completely dead. I put the car on the charger and was able to start it. I decided to leave the car on the charger for about 2-3 hours and then turned it off for the night. In the morning, I decided to try the car again. Car started fine and drove ok. A/C and speedometer still not working however and battery indicator inside the car still lit. As I got to a friend's place, I left the car on as we talked. About 10min later, the car lights started to dim, idle became rough, and then it died. I used a booster battery to start the car but as soon as I turn off the booster battery, the car immediately died. So it looked like the battery may have not been holding a charge. I decided to replace the battery to rule out a bad battery as it was old. After replacing the battery, the car again needed to start with the gas pedal and kept alive with the pedal. I ran a check on the battery with the volt meter and when starting the car the voltage dipped a bit below 12v but after starting never raised more than 12v. I would think it would go to around 14v or so if the alternator was working correctly. So I assume my problem now is the alternator. Do you think that by replacing the alternator, it would solve my car dying out at stop lights, erratic idle at times, and a/c and speedometer/odometer sporadically working? Or do you think there is more problems than just the alternator? I'm trying to decide if its worth fixing given the age of the vehicle.