Am I able to unlock taunts as random or achievement drops?

As far as I'm aware, the only taunts that are in the random drop system are the Schadenfreude and High Five.

They seem to be part of the Hat drop table, meaning that they are rare drops.

The only taunt that is unlockable by an achievement is the Director's Vision, which is unlocked by using the Replay Editor for the first time.

Edit: The TF2 Wiki seems to think that all taunts (except Director's Vision) have a chance of dropping. Personally, I've never seen any of them drop.

If you really want a taunt, your best bet is trading for it.

You don't need yo actually buy things for a premium account. You can sell Steam cards to get store credit and buy things from TF2, and the game considers it a purchase and gives a premium account.