I'm an election official, in Keene. You can show up with the MA ID card. Since you registered late, I'm not sure that your name will be listed on the voter role sheets (I'm not sure what the cutoff date was). If not, you'll go through the new voter registration process, and you'll swear an affidavit that you reside within the town/Ward where you are voting.

Bottom line... If you show up at your correct polling place, you WILL be able to vote. You may need to go through a few extra steps, but the poll workers will walk you through every step of the process.