Vibrates when gas is applied 1959 Chrysler New Yorker

When I drive my Chrysler and increase speed the car vibrates, but when the car reaches speed it stops vibrating.My suspicion was the driveshaft center support, but the car has a whole driveshaft (not split).Could it be the universal joint that needs to be replaced (mine is not greeseable)? I didn't feel anything when I wiggled the driveshaft by hand.