How old are the tires? If they are more than a few years old, that can make the bike shimmy and shake.
The 81 I have with the tires it had on it would headshake bad enough to scare me. I swapped wheels around from my 75 and the 82 parts bike and got it to be ok, not good but ok.
I got a new set of tires on it and it's great now.
To tell how old a tire is read the DOT number, at the end are some numbers in an oval. If the number is 3 digits it is older than 2000, In 2000 they started using 4 digits. The first two digits are the week of manufacture, the second two are the year of manufature.
The bad tires off the 81 were marked 275, as in 27th week of 1995, the other 405, 40th week, 1995. Good tread, no cracking, but no good.
The new ones are 1310 and 1610, 13th and 16th week off 2010.
The tire manufacturers reccomend not running tires over 6 years old. As the tire ages the layer can delaminate, the tread can come off in chunks or as one piece. Bolwouts are common.
A few years back they had a big deal on brand new old tires blowing out on SUV's and causing crashes, a few people got killed.
Companys would buy a wharehose full of tires and take years to sell them. On tv the did some checking at tire stores and found tires dated as old as 15 years being sold.