Hm, getting elected means your value has to be spread far and wide; maybe you should try for something like a Domination victory. Also, if you play on Normal or even Epic speed you might get a better feel for the mod.
As for Heavy ships, they are vulnerable to Fighter, Missile and Quick attacks; I always try to use Missiles and Fast Attack ships (Exploration class) to reduce their hit points before trying to eliminate them. Also, you should always keep Subspace Warheads ready, as they can reduce an enemy fleet to rubble (it makes it really easy to destroy a fleet after they have been hit by a Subspace Warhead). Be careful though not to let one explode on a square next to your Star System.°
Finally, if the Kazon are that annoying to you, you can just select a Custom Game without them as a playable civ - or try playing any one of the various scenarios.
°
EDIT: Personally I think the Romulan Heavy II (Scimitar) is unbeatable, because it can attack when cloaked. So when playing the Romulans, once you can build Scimitars, make a fleet of them and keep them cloaked to intercept any threat - or attack any enemy that can´t see cloaked ships. Even when not playing as Romulans, I always try to keep a mobile force of cloaked ships ready for emergencies.
