You pretty well have only a few options for this problem.
One is the alternator is bad and not charging the battery.
The wiring going to the alternator may be broken/chaffed or have high resistance.
If these 2 things are ok then your PCM is likely at fault. The voltage regulation portion of the charging system is controlled by the PCM.
Sorry, the PCM is the Powertrain Control Module......aka engine computer.
It's mounted under the hood on the right inner fender.
Yes you are correct in your thinking about the alternator.
I suspect you have a wiring issue or a bad PCM.
The PCM's vary a bit in price depending on the application. Probably 6-900 dollars.
You are asking me to guess at replacing an alternator and I cannot suggest that.
Dollar for dollar you might be wise to have a shop diagnose this for you........it would be cheaper than buying an alternator if that is not the problem.