Do the different eras match up with any of the show timelines? I get that this is based on a fictional Star Trek universe from the board game, but wondered if the technology matches up. For example, is the early campaign in the TOS era? Are either of the campaigns in the TNG era? Any guidance here would be appreciated!

It's in the Motion Picture era - between TMP and The Undiscovered Country, I believe.

As in all three Starfleet Command eras from the game span this time?

They are set in the TMP era so SFC 1 2 and OP are all TMP era but in their own timeline separate from canon

SFC 3 is set just before ST Nemesis.