Well it is definitely electrical, since you just changed alternator is the belt tight and the connections correct and tight.
I say this because your symptoms go along many lines, here is one. Car drives to store but battery is not charging, car is running on battery power only. Once you get out and back in battery now dead and weak, so you jump it, that would work as battery now provides a punch of power....it fires and runs for a few minutes, but when you put it in drive and add more load.....it dies.
So that could fit, hook multimeter to rear of alternator and see what it is putting out.....14.2 would be good....slipping belt, bad alternator are possibilities if not.
Does it have an external voltage regulator? On the firewall? That must be good and hooked up as well
Also a shorting neutral safety switch, or high resistance in starter wire, or even a loose or corroded battery ground...all need to be clean and tight
Look for bad or corroded battery connections or ground...check alaternator output and see where you get
You will get it!