Balance Druid vs Shadow Priest

I'm at a loss on how to fight Shadow Priests.

Their CC is better than mine - they can dot and start damage while I'm feared, I can't do anything but heal myself while they're hurricaned.
Their dots are better than mine. They can also dispel mine and I can't dispel theirs.
Their healing is faster than mine.
Their shield is better than my barkskin and comes off cooldown much more often.
Their damage face to face is better and I'm getting slowed at the same time.

The rare time I see success is by popping trinket, pop trees, pop starfall and try to burst while looking for a heal cast to interrupt with a typhoon. This works once in a while, but experienced priests will disperse through the damage or run behind a pillar. Then I've got nothing left.

Attrition mostly. As long as you got enough resilience and mana return, you're fine.

When getting dispelled while healing, swap to lifeblooms rather than plain rejuvs.
Needless to say, protect your Innervate.

Starfall early to force their CC or dispersion, you can now turtle and use cyclone offensively (as an interrupt for Vampiric Touch/Flash heal) until your next Starfall. If you're behind on damage, just cyclone and start over. He has full HoTs and PW:Shield, just cyclone and start over. You're just waiting for next Starfall.

Your win condition is catching him wasting CCs and Dispersion + you have Starfall up + he OOMs faster than you. If you're still lacking damage, don't worry too much about refreshing moonfire/insect swarm, just hardcast Wrath/Starfire more.

Other subtle details is obviously try to pull him away from pillars to get the maximum value out of Starfalls

More practice!!! All the best.

Lifeblooms heal you when dispelled.
Do everything you can to starve them of mana, cyclone dispersions, cc shadowfiend, typhoon prayers.
Spam low mana buffs to waste their dispels, coz dispels cost lots of mana.
In the end whichever side screws up first dies.
I would just run away, it's no big deal to tank their dots with hots & fly. Much more annoying for them to land, kill mobs first if they aggro, mount up, and try to chase you.