I think your last line says it all. Budget "I'd like to try and stay under $6000"
Unless you go full on junkyard commando, Driveway Engineer (YouTube guy) and get everything used and do all the work yourself, you'll be hard pressed to get an LS swap done for under $6k. Plus it'll take you probably months at a minimum to have a vehicle you can trust to drive regularly. That's because there will be 1001 little things you didn't plan on and couldn't account for that you'll have to take care of.
Granted, if I won the Lotto tomorrow, I'd be working on an LS swap the day after that. But only because I can hire out most of the work and throw money at the problems.
The 383 swap on the other hand is essentially painless and can be done very inexpensively. I know because I've done it. Check my sig.
I don't see what year your truck is mentioned in the OP, but if it's a late model with port injection and you haven't done the CSFI to MPFI conversion, do it!!
In my case I bought the whole core 383 package which meant I got a 4 bolt main, forged crank, roller lifters and "RV" cam, hypereutectic pistons, and other parts that I would have wanted anyway because I my case I was as much interested in increased durability as I was a power increase. In your case I'd recommend the same. At least get a 4 bolt block and upgraded, matching crank, rods, and pistons. A really should do is add a decent cam that boosts low end torque and gives a little power too. You'll definitely need to increase breathing so a cold air intake and headers with a good exhaust are a must. Finally you need to custom tune the PCM or most of this is wasted. If you're going to do an 0411 upgrade, now you're starting to push the complexity to the point where you might as well start considering the LS swap.
Anyway, back to the 383. A quality built 383 with the right components can be powerful, rugged, and reliable and is IMO still the best street SBC for big cars and trucks. They can make power with the best of the 5.3l LS's for a lot less work.
Having been in this position and having faced the same choices, I wish you luck. It's not an easy decision. We're lucky to have a platform with so many good options. Whatever you do, PLEASE post it and all the things you faced along the way so everyone else can benefit from your experience.
Personally I've never regretted my 383 upgrade for a minute. I've only ever regretted not having the money to do more and better things to it.
Like move to an aftermarket EFI system and bigger cam.