Rear differential swap

The short answer is no. The P/U trucks use a different axle design than the Avalanche Tahoe and Suburban. Width and mounting are different.

Can I take the cover off and take out my open diff and put the limited slip from the 2003 in?

That would depend on the gear ratios in the two units as there were two differential carriers made, if the gear ratios match then you could swap the carrier in using your original gears, if the Sierra has the 3.43 or 3.73 gears and the other has the 4.10 gears then the carrier is different and wont fit. The other issue would be if one is a 2500 while the other is a 1500, that can get interesting depending on it's actual axles, GM used both a semi-floating and full floating axle in those years and those are different internally.
You will also need to check the gear engagement pattern even using the old gears, don't skip that step please.

RPO codes for gear ratios, the sticker should be on the glove box door.
GT4--3.73
GT5--4.10
GU6--3.42
G80--locker

Okay, they are both 1500 so we will see if they are the same ratio. I'll let you know. I hope I can have a limited slip differential.

The G80 isn't a real limited slip it is a modified locker that uses clutches instead of a ratchet or gearing locks. I don't like the G80 personally because in real off road use they have a bad habit of grenading. If you are not beating it up a lot then they are okay.

Thank you for telling me that because I go off-road a lot.

The way they work is strange. They are like a limited slip with clutch packs that stay loose until one wheel starts to spin. At that point a weight starts to fly out of the unit and starts squeezing the plates. If it keeps spinning the weight pops out farther and locks the unit up. As soon as both tires are turning the same speed the weight starts to retract. That releases the clutch pack. For street use in snow or sand it works okay most of the time. But if you are using it a lot the pin and weight start to wear and bang, the weight lets go, then some of the other parts come free. The problem is that there isn't enough room in the case for those parts to just fall out of the way and the thing comes apart. I've seen parts get between the ring and pinion and snap the nose of the housing, get under the ring gear and break out the bottom or a few where the weight came free and exited out through the back cover. I had a conversion van with one in it and one of the first things that I did was toss it out and installed a Torsen T2 unit. I thought about something like a Detroit or air locker but wanted something that I knew worked and didn't chew up tires or require extra steps to work as the wife sometimes drove it.
Now if you run off road a lot and stay on dirt or loose surfaces and don't mind extra tire wear, a Detroit is hard to beat. They can be touchy in rain or snow though.

I'm just hoping they have the same gear ratio so I can swap them. Because I really need a limited slip differential, but I don't have much money.

There are probably a few videos out there about them and what to look for as far as wear and damage and how to do the swap.

Okay. Thank you.

You're welcome, just be real careful about installing the gears correctly, you can end up with a noisy gear-set or worse a quiet set that grinds itself to death in short order.

And if I do it right there should be no bad noises right? lol

Hopefully, although I have seen units that ran real quiet that still chewed themselves up, usually because someone mixed parts, like the ring from one and the pinion from another or installing a new ring with a used pinion.
Keep us in the loop please, it's nice to actually get a conclusion one way or the other.

So use all the parts from one differential, no mix matching?

You will need the ring gear and pinion to be matched. Same with the bearings for the carrier and pinion, The outer races will wear with the bearings so they need to match. Plus a new crush sleeve to set the proper bearing preload. Shim wise, write down the shims that are on the G80 in the old unit, that can be a starting point.