I do not know if you need an alternator. Let me summarize my deal. Maybe it will help you.

Car towed home. Battery was very dry. Pulled voltage regulator which showed wear but not a tremendous amount (see pictures in this thread).

I filled my battery and put the old voltage regulator back in. It read around 12.0 volts not started. Then started it was around 13.7 at times but then fluctuated in to the 11's at times. That scared me. So I changed out the voltage regulator. Now the battery cold has been giving me readings of 12.45 and when started at idle it reads 13.7-14.1. I have not had a problem since.

Have you checked your voltage regulator? They sell for around $29 online plus $5.50 shipping.

Because you never checked the alternator first ,it's hard to say what was actually wrong,but i would put money on a sick battery.
A car battery should read 13.5 to 13.8 volts cold.
There are six cells ,each produce 2.1 to 2.7 volts.
11 volts means one cell is sick ,probably shorting because of sediment from cells that have been run dry. When this happens,the cells shed particles which build up in the bottom and bridge the plates. You will get 13+ volts with the engine running because that is alternator charging and it's not the battery condition.
A quick look into the cells with the engine running will show the bad cell,it will be bubbling more than the others.
The best idea is to drive to a place that sells batteries and get them to do a load test. This will show you that your battery is due to die soon because it's not keeping 13 volts when you stop the engine.