Yes, it is common for tire clearance to be very tight on road bikes from the 1990s-mid 2000's, and they may only take 23mm tires. That was the fashion then.

Wheel 'tuning' or Truing, as it's known, is fairly straightforward once you know what you're doing. It does require some simple tools and a methodical approach. You don't just 'tighten all the spokes' as it can lead to some unintended results, as you found out.
It's not a difficult thing, but there is some nuance and patience required. The tight clearances of a sporting / racing road bike like your Lemond are not a very forgiving field to learn on, either.