It's worth noting that Weld lost the 1996 Senate election (to Kerry) by 7% when he was the sitting governor.
Kerry won in 1996, because of two things...
1. Clinton was running for re-election and won 60% (or so) of the vote.
2. Weld's LT Governor: A. Paul Cellucci
If these two factors were not in play, I'm convinced, Weld would have won.
I recall when that election happened (I was in MA and a junior in HS). The reason why many voted for Kerry over Weld was because many had doubts over Weld's LT Governor taking over if Weld won. Cellucci (Weld's LT Governor) has controversey over his personal finances (he had a huge personal debt), and many did not want to see him as Governor.
Oddly enough, Cellucci became governor in the summer of '97 when Weld resgined. I'm convinced Weld didn't really resign to fight Helms over his appointment - that was a dead duck. I think he was trying to set Cellucci up for the governors race and it worked as Cellucci won in '98.