You won't be able to even come close to an LS swap with $5K. I've seen articles on hot rod sites that say: "$5,000 junkyard LS swap! Done in my garage with hand tools in one weekend!"
That's not even close to reality. When you do a complete swap like this, there are always things that you didn't anticipate that come up. (Where will you source the motor? Which transmission do you want? Which transfer case will you use? How will you keep the engine cool? Where will you put the transmission cooler? Do you want a/c? What about programming and the wiring harness? Will everything fit? etc.)
If you want to know how I know this, I'm at the tail end of an LS swap myself. And project creep is a real thing. Like you, I have kids. I did some of the work myself, but I ended up farming out some of it to a shop too because I can't spend every single weekend working on my truck. And that's a quick way to burn through piles of cash when things you didn't expect come up. Just ask my wife if my project stuck to the original budget.
Fortunately, I'm married to a very cool, supportive woman.
If it's a weekend driver, I would just drive it for what it is with the 360. It'll never be "fast," but that's not really the point with FSJs. Put an aluminum intake on it and an exhaust. (You could do headers, but I didn't think they were worth the trouble on my previous rig.) A new carburetor will help too, but a lot of people (me included) go too big when they are building a 360 for the first time. A hotter ignition system will help too, and it'll leave you with cash left over.
Some other thoughts:
If you swap out the cam, you will wake up the 360 a bit. But you will also run the risk of flattening a lobe during break-in.
Fuel injection would make cold startups easier and completely get rid of carb adjustments — but it won't gain you any fuel economy in my experience. But then you'll have to deal with a new fuel system or a surge tank to eliminate fuel cavitation and fuel sloshing around in the original (non-baffled) tank. (Ask me how I know.)
Start with a compression test and consider how many miles are on your engine before you spend too much cash waking it up. I've even heard of guys getting into their engines with borescopes to look at the cylinder walls without taking the motor apart.
Everything will e said above is sound advice. If you've read this far, go back and reread his post.