Ah, a problem as old as Dungeons in MMOs.
Just get a friend who wants to level a healer/tank and you get your insta invites as a DPS.

Also, the wait time for a dps is not really that long and in case of trails, alliance raids and normal raids often even shorter then tank and healer queues.

That's why I level my DPS classes by doing these and choose tank healer for the leveling and 50/60/70/80 roulettes.

Or, do something else while waiting.
Normal quests, fates, glamour, RP, crafting, collecting, beast tribes, etc.
When you play smart you will have no wait time.