Battery lite stays on when engine is running

This light on dash is telling you, alternator is not charging or current is not reaching battery. With a voltmeter check your alternator output at battery. It should be at least 13.5 volts. If less than that check output at back of alternator, big red wire bolted to alternator. If voltage is within specs. then you may either have a blown fuse, open(broken) big red wire from alternator to baterry. Check big red wire circuit integrity by proving back of alternator where wire bolts to alternator. With the engine not running, there should be battery voltage at that bolt/wire. If no voltage repair open in circuit, either a fuse or broken wire( usually). If still no output(13.5 volts) or more, check power ground to alternator plug. If these power ground check out ok, then you more than likely have a defective alternator.

When the warning light is on the alternator is usually not charging the battery. Could be a wiring problem.

Sounds like a charging system problem. The engine runs off of the alternator but if it is not charging the battery properly the battery will be drained until I engine will no longer run. You need to get the vehicle to a repair shop to have the battery and charging system check for defects

open or shorted alternator circuit (control wire)
let your auto store test the alternator charging
on and off the truck.
inspect 3 wire connector for burn or lose connection
disconnect 3 pin wire and start the engine.if the light stay on you have shot to ground.

You put a new battery, now its acting like very low power again?
Most likely the alternator is bad. Maybe not putting out enough to recharge battery. Could have been the original problem.

Can be checked with a meter. Battery at rest should show 12 - 12.9 volts more or less.
With engine running, should be showing 13.5 - 14.7 volts as alternator is pushing electric back into battery.

If you start engine and (after say 10 seconds) voltage stays same or drops, then the charger needs to be replaced. Thats the alternator. Shop around at different auto parts store. (Phone first)

need to test and replace alternator

Its probably the alternator. It will need to be tested on the vehicle with the engine running and the light illuminated. Measure the battery voltage with the engine running. It should be at least 13.5 volts. If it is lower have the alternator tested at a shop with a charging system tester.

It is telling you your alternator is not charging properly. This could be due to a loose belt, poor connections at the alternator and/or battery, or perhaps you do have a bad alternator. Check the voltage output at the battery, with the engine running. It should be 13 to 15 volts if the alternator is working correctly (depends on engine speed and battery state). A good auto parts store, or an auto electric shop can test easily, and many times do not charge except for repairs they actually do. Regardless, you need this fixed.

I once replaced my alternator in a S10 three times before I got a good one that put out 14 volts, this one came with a lifetime warranty and cost more than the others. Chevy's used to have a problem with a rebuilt supplier. Have you checked for a bad ground wire? Try one more time but, make them test it before you pay and install it.

If you live in the US, take it to an Auto Zone store and they will read the code off for you, this may be true of other major auto parts outlets, as they hope to get your parts business.
Once you have the code, you can decide the best course of action.

Hope this is of some help.
Ed