Either one could be correct:
The contract will not be signed until you have checked the manuscript
and [have] confirmed that everything is ok.
or
The contract will not be signed until you have checked the manuscript
and [then/thereafter] confirm that everything is ok.
I would say they could even be used interchangeably. The distinction in meaning from use of one tense or the other is negligible. Given that both things have to happen before the contract is signed, and that confirming would happen near immediately following checking, it makes no difference whether one says "check and confirm, and then we'll sign" or "check, then confirm, then we'll sign".