(some things that I try to do, don't take as gospel)
For trash you'll want to try to use your CDs towards the start of a pull (Barkskin, Frenzied Regen, Incarn, Survival Instincts if you've got a spare charge and know it'll be safe) with the goal being to stay up at full health for as long as possible. Keep your rage floating up high so you can roll double stacks of Ironfur after your CDs are wearing out. Once your health starts consistently dipping under like 60-70% you will want to start kiting and get a reset on your stacks. Assuming (for now) that you only have yourself, you'll want to Typhoon and then either a kitty sprint or Stampeding Roar. If you take Wild Charge you could drop bear form and leap to a range to cover extra ground. There are also many instances where you can use Travel Form and just leg it.
There are many mobs scattered throughout the dungeons which cannot be slowed or knocked back. For those packs you're going to want to start kiting early, probably stick around just long enough to get some decent threat and then do your knockback. This way you'll only only be getting stacks from one mob and should be alright. Those mobs are usually focus targets as well so it shouldn't live too long and if your group's DPS is good enough it could die around the same time as you would want to kite anyway, so just group and nuke.
Some traits you could look into if you need more help with trash:
Strength in Numbers - The shield should proc on pull giving you bit of a buffer at the start while your other stuff.
Bulwark of the Masses - Either refreshing between pulls (shield is up for pull) or at some time while you're kiting (help negate that ticking damage at high stacks).
Classes you can look for to help with trash:
Balance Druids - Get them to hold Treants for you rather than using it off CD for DPS. Coordinate using it around the middle of a pull and you can get up to like 30 stacks (provided health is safe) and then reset stacks using the AE taunt.
Frost Mages - The damage they deal just passively (I think) applies a slow to all the mobs, when you need to kite you can literally just walk away from the mobs and they'll be heavily slowed.
Hunters - Have them drop a Tar Trap behind you to help slow mobs when you're kiting. You can then save your Typhoon for after that for a greater window of no damage, low on space, etc.
Monks - Same thing as Tar Trap only they have Ring of Peace which will spam knock back on the mobs trying to run through.
Warlock - If you know a pull is going to be particularly savage you can get them to put a Gateway Down before pull giving you a big leap up your sleeve.
Shaman - They can drop an Earth Elemental once every 10 minutes which AE taunts everything and lasts for ages.
Restoration Druid - They can drop an Ursol's Vortex on top of you when you're going to kite which will grip all the mobs back one time. You can then Typhoon after than.
For those big mobs which are immune to most of these, you can use a variety of classes to taunt for you and kite/tank to get a stack reset. Druids, Warriors, Demon Hunters are pretty good options because they all have their various ways to kite or do something so they don't die.
Now for Bosses. Most bosses will have some window of time which will just naturally allow your stacks to drop off. There may be some times where you get stuck on high stacks with no real option. In those situations you may have to get a DPS or healer to taunt the boss for you. If you can, make distance between yourself and the person, and then have them taunt. With a bit of good timing (dodging some slow boss swings could have the stacks 1/2 dropped already) you can taunt bounce the boss back before it ever hits the person and have the stacks reset as the boss runs back to you. Also some classes can just face tank the boss for the 6 or so seconds required. Demon Hunters with Blur and Metamorphosis up (or Netherwalk if they take it) can hold a boss fine. Myself as a Balance main, I've taunted the 3rd boss in a ToS+13 and just face tanked with Barkskin while the tank's stacks reset and survived.
The only trait that could help a little with bosses is Winds of War for the extra dodge. I think I have 1 of these and between slow boss swings and a good dodge chain stacks just tend to drop off pretty often.
Hope that's all given you maybe a couple of ideas to try out and see how it goes.