I play a holy Paladin, and I've had ZERO problems ever finding groups. Took a break and played an enhance shaman on the horde....ended up having to pvp just to get out of greens and had to respec resto just to get my karazhan attunement. Impossible to find a group. At all. HUGE difference.
Overall there is never a shortage of dps classes when trying to put together a group. However, there is very little glory in healing and you usually get blamed for every wipe, so both damage classes and healers have their tradeoffs.
You can skip both damage and healing and go tanking, which has even more demand then healers. The LFG channel is basically the LFT channel (looking for tank). There is no faster way to get in parties, guilds and endgame content then being a tank.
-VGsatomi