Do they even make thread chasers for internal tapered threads? Don't recall ever seeing such.
My first step would be to liberally lubricate the plug with teflon pipe dope (which is what I also use as 'sealer'), and put it in straight. If you can get it started straight, it should take care of the cross threading.
If that doesn't do, I'd try tapping it just a bit deeper with an ordinary NPT tap, preferably an old one. But don't go too deep, as it is possible to cut the hole oversize until the plug won't fit.
If that won't work or you don't have a suitable tap, check your FLAPS for a "repair" drain plug with oversize, self-tapping threads. Unfortunately it becomes a permanent bodge until you can get it apart and install a Helicoil or have the hole welded up and redrilled.
Oops, ignore that last bit. Apparently no one has them in 3/8 NPT. Just buy the tap, it should be readily available. You'll need a tap handle or socket and T-bar to fit it as well.