There is a general lack of continuity between how fervently Olivia wanted to make up for election rigging by having Grant be elected fairly the second time around, and how okay she later is with going to great lengths to cover up genuine problems with his administration--like releasing Eli from jail to cover up Grant going to war (and causing deaths of U.S. soldiers and Sudanese civilians, then lying to Congress about it) to save her.
The fact that Abby so quickly heard rumors about the Senate Judiciary members being blackmailed shows the weakness of this plot. If the Senate suspended its impeachment proceedings immediately after Senators were, on television, handed envelopes, they would certainly be under suspicion and have their own actions investigated. And although the show writers are having pundits and news reporters stating that this was a win for the White House, Grant's statement that "Congress has fulfilled its constitutional duty, completing their fair and thorough investigation into my administration" would certainly bring down a firestorm of criticism given that the Senate didn't complete an investigation, but canceled it under suspicious circumstances right before important witnesses were called and right after the then-First Lady was discovered to be lying to Congress (about Jeannine Locke). In general, show writers simplify politics by downplaying partisan opposition.
No reason is ever given as to how Fitz understands the Senate calling off its impeachment hearings. One moment he tries to marry his mistress to prevent disastrous testimony, the next moment the whole thing is called off and he doesn't seem to question why.