In my quest to recycle, I've been using the bags to empty our cat's litter box, but the number of bags we get weekly fars exceeds our cat's ability to poop. It seems like the volume of plastic could be turned into something useful, even if it's just a lump of plastic to use for a paperweight.

While your goal is laudible, I am not sure that you will really be "saving the environment" by doing your own plastics recycling. The problem is that you can't achieve economies of scale, which are present in other things than mere money--energy, for example. The method of using argon that I outline below is rather wasteful of argon, which takes energy to isolate from air (though common in small-scale operations such as research labs).

The best solution is for you to set aside a space for the stuff, and periodically cart it to the local recycling center on your way to somewhere else. That's what my family does.

But to answer your more specific question:

You can melt plastic at home, without charring it, the same way it's done in industry: by using an inert gas blanket and careful heating. You can buy argon in tanks, from industrial gas suppliers; argon is heavier than air, so if you use a fairly deep vessel to melt the plastic you won't lose too much.

The procedure would be this: Segregate and shred the plastic. Place the shreds, one type of plastic at a time, in a fairly deep, heat-resistant vessel (a saucepan rather than a skillet), and run a stream of argon into the vessel for a time. Then, using a hot plate or other controllable heat source, gently heat the plastic until it melts while continuing to run a stream of argon over it.

"A time" and "gently heat" are parameters you'll have to play with, I'm afraid. But as long as you do this in a well-ventilated area, with a fire extinguisher handy, you should be in no danger. And it may be that you will find some plastics that you can melt without charring them.

You will need to use some ingenuity in setting up your apparatus; you will also need to buy a regulator and a safety harness for the argon tank, and some metal tubing and gas-tight fittings for your argon line.

If this sounds complicated, it's probably a good argument for leaving plastic recycling to the professionals.

Incidentally, plastic is not always melted for recycling; sometimes it's dissolved in a solvent, then spun into fibers. But I don't think you will have the appropriate equipment for that at home!