2000 Chevy Tahoe Tahoe Sputters

When I start my Tahoe the engine will like sputter. Sometimes it will die out other times it will not. Once I get started it drives fine. Other than when I will come to a stop sometimes it will just die out and the battery light will come on. I normally just put it in Neutral start it back up give it a pump on the gas, apply the brakes and I take off. After driving it for a while it normally run very smoothly. My husband fines that if you turn the A/C's off it doesn't do it as much. I find no difference. I did find that it tended to do it more so when I was low on gas, now it does it when low on gas and when I fill up. My gas gauge also started where it will drop all the way to empty even when it is full. Also after filling up It takes about 5-10 mins for the gas gauge to go from empty to full. I have put it on a machine to see why when it is first started it doesn't idle smoothly it acts like it wants to die down but it does not register anything. I love my tahoe and would love to have it fixed.