The correct fix is new tires.
If you wont do that, the next best is to tube them. I don't know what kind of nonsense you're talking about with punctures. How many punctures do you honestly think you'll get between now and when you replace the tires(which you still need to do, the tubes are just a bandaid for now)? Even if you get a puncture, patching a tube is easier than patching a tire, plugging a tire may be about equal. Tubes will probably run you half the cost of just replacing the tires though.
If you refuse to buy tubes, go ahead and put slime in it. You might just end up wasting money though. I know everyone here loves slime but I see that crap come in inside of tires all the time. Why are the tires here? Oh, because they still leak. It's also pretty nasty to the rim if left in there for a extended period of time, will rot it out quicker.