It’s probably not the battery because you should hypothetically be able to run without a battery once it’s started. I would guess that the alternator might be bad even though it tested out ok but there could also be some type of crazy power drain (or short) somewhere. One way to be sure of this is to run this test at night. Turn the headlights on and as the car starts to stall do the headlights dim? I would shine the lights against a wood fence or house so it’s easier to tell. If so, it’s probably an electrical problem. If not, you may have some other type of problem. Does this always happen or did this just start? Did you run completely out of gas recently? Every time you run out of gas you suck up all the debris from the bottom of your fuel tank and shoot it down the fuel line. If this is the case, it would also be prudent to change your fuel filter. Maybe your fuel pump is starting to go bad. There’s a ton of stuff it ‘could’ be on an older car. Test the headlights to see if it’s electrical, then we’ll go from there.