It isn't your balance - it's the bike. The bike you rode when you were a kid had laid back seat tube and head tube angles - approximately 70 to 72 degrees from horizontal. The Lemond you're now riding has probably 73 to 74 degree angles. These angles used to be considered "racing only" angles because of their twitchy handling. The more upright angles (and the more twitchy handling) has, sadly, become standard on almost every current "road" bicycle made.

With practice, you can sit upright on even today's bikes, but it isn't as easy. Practice, practice, practice. I can ride both of my bikes without hands, including going around corners. A bit more speed helps smooth out the wobble.

Also, the "longer" your saddle (from front to back) the more control you can exert with your thighs while upright.

Bicycles can still be bought that have geometry like your old Schwinn, but such "custom builds" aren't cheap. Your easiest option might be to buy a used Schwinn on e-Bay & bring back the good old days!

Good luck with it, and yes, it IS nice to be able to sit up & stretch for a while whilst riding!