Unfortunately it's not likely to be a cable adjustment. But, obviously it would be best to check that first as it is pretty easy to do. Even without the centering tool, you can remove the airbox and get a better look at the transaxle linkage/cables. You can have someone sit in the car (engine off) while you look at the linkage/cables from the passenger side. Have the person inside perform several 2nd to 3rd shifts and make sure the cable length isn't hampering it from getting into 3rd. As I said, more than likely it is an internal issue (sorry!) - but check the cable first.
If not the cable, you might keep an eye out for anyone selling a transaxle. If I was in your shoes, I'd try to acquire a known-good transaxle and then have someone swap it in. That will give you some time to have the transaxle disassembled, diagnosed and fixed. Once good, you can sell one of them.