My swap is done and my girlfriend is still waiting for a ring after 9yrs today.....I guess I got my priorities right
On topic tho; the 5.3, tho smaller than the 6.0 is still a top-notch motor, and totally accepted as a performance and swap engine. Its not like getting a smog 305 instead of a 400, its more like getting a 327 instead of a 400. (DISCLAIMER, that comparison is not mathematically correct, but provides the correct message).
The 5.3 is readily available with a 4l60e already behind it, this benefits you in 2 ways. First, you don't have to track down a 6.0 and 4l80e together (since many 6.0 trucks were with 4l60es, this would mean you'd have to locate and purchase a seperate trans), and the obvious cost savings of buying as a package.
The 5.3 with a cam package will sit comfortably in the 330-370whp range, assuming all the supporting parts are in good order. This is well over what the 350 would be at, and doing so through an overdrive tranny, giving you averages from 15-21 mpg (lots of factors in that).
Your budget is a bit tight, but I think a stock-parts 5.3 swap is doable under 3-4k. If you plan, re-do your own harness, and don't have to do any rebuilding it shouldn't be terribly difficult to do so. Things like fittings add up fast, so planning on the fuel system, trans plumbing etc is very helpful in stayin on budget.
You'll need to have the PCM sent out or tuned locally to remove the VATS and emissions components. Learning to tune is awesome, but is better timed AFTER the swap is running. You'll end up frustrated if you have to worry about the mechanicals of the swap, in addition to teaching yourself tuning, plus trying to keep your now-legally-entitled to half your stuff significant other happy.