Be more tactical!
Well, I don't know what to say here. I take pretty long time playing through maps in CTB as well, but considerably faster than STB. I guess, keep your team small, and try to place them in position before they engage in combat. Use a lot of cover. If you have a team of three (Stitch, Farsight and you) set two guys to give covering fire (Farsight a bit further away as a sniper and Stitch closer to you with his shotgun) to your main guy when you get ready to enter a new building. Place yourself and Stitch by the door, then enter with one of them, crouched so Farsight can shoot over you. Or try and lure people out of the building to recieve a shotgun surprise to the face from Stitch when they get out of the door, just run in with your main, burst the closest guy, then sprint back behind Stitch. Really, just think tactical, think several steps ahead, keep your squad small, place them behind cover a lot, before they engage enemies, and to cover eachother. If detected by enemies, well, get them behind cover, and try to work out a solution. If the enemies are too far away, keep the close-ranged people behind the cover while trying to get in a good position with the sniper. Or try and find a good route for one of your close-range people to sprint to the enemies side and flank them. Dangerous tactic if there's many enemies, but works good sometimes. Or you do both, have the sniper distract them while flanking with the others.
And you will have to eat through healing items in Tactics
Those raiders get lucky shots all the time, that's why it's good to have Stitch, the medic, around.
Hope someone more hardcoe Tactics player can show up and give you some better advice!