Yes the 4L60/65/70E's use the same wiring connections.
I recommend going with the 80E transmission. There is a detailed write up on the swap in my GMT900 thread (linked in my signature). A GMT800 will be similar and the information is out there. It does require a segment swap on the tune to run the transmission. I'm guessing an 05 5.3 is Gen III architecture (knock sensors in the valley cover, cam sensor in the rear of the block, 24X crank trigger wheel, etc.)? Is it DBW or DBC? Somewhere around 04 or 05 was the crossover years, if I recall correctly.
As long as you go 24x-->24x or 58x-->58x, DBC-->DBC or DBW-->DBW its a direct swap from the 5.3 to the 6.0. But I would still flash your PCM with the tune for the 6.0 or a custom tune to get the most out of it. That is, unless if you are going to replace your PCM with the one for the 6.0. There are a lot of ways to overcome any mismatch. For example, in my brother's 78 Z28 we have a Gen IV 6.0 but are using a Gen III PCM because we wanted to run a DBC throttle body. That necessitated using the LIngenfelter converter box to translate the 58x crank signal of the Gen IV engine to the Gen III PCM which is looking for 24x. Also, extension harnesses are available for when you put a Gen IV motor in a Gen III car and need to extend the harness for the cam sensor up to the front cover (where it is on Gen IV motors). Another example is there are stand alone transmission controllers for the 80E.
The closer you come to finding a 6.0 that is equipped exactly like your 5.3 (engine generation, throttle body type, PCM generation) the easier and less expensive the swap will be.
When you find an engine you are serious about, ask some questions. There are lots of guys on here that are LS knowledgeable.