If I'm worried about my body heat, I will usually die when entering Gazluk to reset my heat to max, then sprint for the cave I want. The New Prestonbule cave has friendly NPCs (including vendors and a bit of storage), the Tower View Cave is the easiest cave to start farming for gear.
Both caves may have some mobs hanging around near the entrance; don't stop to fight, just zone in asap. I find my key bindings really helpful here if the cave entrance is covered in nasties: I'll spam my key for "Select Next Non-Combatant" while running until I have the cave entrance selected, then I run up to the cave and hit "Use Selected" to zone in - much smoother if there's no mouse target and clicking involved. I'd recommend being very close to the entrance before hitting the Use button - the AI pathing near the entrances can be a bit erratic and you don't want it to slow you down to get chewed up by the six mobs that have been chasing you.
Note also that the tower opposite the Tower View cave is both warm and a safe zone. Mobs will not follow you inside, so you should be able to stand in the tower and lose aggro. I've had occasional problems with range attack mobs not resetting right away, but they should lose interest in you eventually.
Once you are more comfortable in the zone, you can use the NPC Outpost in the northeast to buy some racial slot jewelry that will allow you to stand near glowy yellow crystals without getting sick. The crystals give off heat, so they are a great spot to warm up if you are too cheap to buy firewood.
Oh, and make sure you have food buffs going when running around outside - it helps a lot if something starts hitting you or you are taking frostbite damage but are almost close enough to reach a warm spot.