Bob, I think Stretch has you going in the right direction on this. I had a Lario from the same era, 1986, that did the same thing. I ended up replacing the ignition switch and that was the end of the problem. However, IIRC, you can remove the ignition switch, carefully pry the case apart, then clean up the contacts with fine emery paper and have a similar result to a replacement switch if the contacts are not corroded or burned up too bad.
BTW, the T-5 is a really nifty bike. I had a chance to ride one once and really enjoyed it, but I guess I have a thing about 16 inch wheels. (hence, multiple Larios...)