Don't bother with the Adjudicator.  The TEC Ogrov can work if you use it right, but the Adjudicator is more expensive and deals less damage, so it's just not worth your time.

The starbase itself actually isn't hard for Advent to beat.  Bring some Iconus Guardians and Progenitor Motherships to boost your shield power and you can tank right through it with enough destras.  The big problem is the Vasari fleet backing it up, which likely has lots of strike craft, support cruisers, and firepower.  That's quite a bit more challenging to take on, particularly if they have lots of phase missile research.

Generally speaking if we're talking medium to small fleets, you won't be able to capture heavily fortified and defended worlds.  Focus on attacking other locations instead or building up to a bigger fleet level.  Once you break the 1000 command level, Vasari starbases start to become less intimidating, and if you approach the 2000 command level they are by far the weakest starbase because their damage stops scaling after the early game (unlike TEC and Advent varieties that offer uncapped area of effect support, damage, and disruption).

True; if he's got a clump of hangers and repair bays together, the Adjudicator would work quite well.  If they're spaced out, however, I'd be more reserved, and if the enemy fleet is there... I'd back off.

You need to proactively use repulse to keep assailants out of range of your Progenitors, and actively use telekinetic push to keep bombers away.  Advent is very dependent on repulsion to win its fights, particularly in the vs Vasari matchup.  Against phase missiles, it's the only way to keep your capital ships alive, and you need your capital ship abilities to make your fleet competitive.