Do this via level select, rather than signing a new pact.
Trauma difficulty is unlocked after you have beat the game on "Nightmare" difficulty and for the most part, is no different. The main things to remember are...
1) Souls no longer give you health, so you are going to have to rely on the yellow souls to give you life back. The green souls still charge you towards "Demon mode" which makes you invincible and incredibly deadly for a short period of time.
2) Collect all the ammo you can. Enemies still die easy, but you don't want to get caught out with a bad weapon for the wrong enemy. For instance, the rocket launcher is great for taking out Big guys and groups of enemies. The stake-gun is great for taking out enemies that are further away.
3) Acquire all the tarot cards. These give you achievements and make the "Trauma" difficulty so much easier. The best set up is ...
GOLD: 4x Damage, Cards last 20 seconds longer, Iron Will (No damage)
Silver: Divine intervention, Mercy
This set up basically makes you a titanic super-god who can't die while the tarot is active. Use it only when you are getting swarmed such as in The Colosseum level at the end. The mercy card allows you to use the cards 3 times each level, which is a huge help, drastically making the game easier.
Other than that there isn't much more advice to give you. Be on the lookout for secret areas as well as these generally give you more ammo and health/armour which is always a nice bonus. Always keep moving, preferably bunny hop as most enemies cannot keep up with you. Use explosive barrels to take out groups of clustered enemies.
Good luck, Pm me if you have any concerns or worries.