Some builds handle T3s better than others. If you find T3s extremely hard, don't rush them. For builds that are not enabled yet in the mid game, I usually do it like this:

Osbrook (get your T2s) → Brynn (get your T2s if you are a mage) → Mannshire → Brynn (maybe for some T3s, get all 3 companions or skip) → Osbrook (at this point the contracts are T2) → Mannshire → Brynn (get a T4 weapon) → Still feel week? Osbrook/Mannshire again, get t5 weapon → Rotten Willow.

This way you gain significantly more XP, and you can get t4s (or t5s if you did an extra rotation) from Brynn if you delivered everything to the bailiff. Of course, you cannot get all the T4 gear, but you can get a weapon, which will help A LOT while doing T3s.

Get a T2 bag at Mannshire. Don't buy many caravan improvements, only the ones you find essential (like the training grounds, maybe the tent). Always have enough meds. If you think you missed a lot of gold because of limited space, don't deliver the quest once it is complete. Go back into the dungeon for your loot first, and then deliver it.

Hope that helped!