Is it possible to just choose to get rid of one of your own creatures, just because you want to, without card effects that specify sacrifice? I can't find this in the rulebook. Does anyone know of any rulings?

For example, lets say I have Hastur The Unspeakable in play, and Hastur The King In Yellow in my hand. Can I arbitrarily get rid of Hastur The Unspeakable somehow, so that I can then play the other Hastur from my hand?

Afraid not. Without some other card or rule effect you are stuck with what you have on the table.

I look to the cards for easy things like this. Look at "Living Mummy" for example. imagine if you could just discard it whenever you wanted. When ever it goes to the discard pile, you wound all characters. that could be huge for clearing the board early. - food for thought

Best to decide which version of Hastur works better in your deck and go with that one only.