Is it possible to control or disallow giveaways from others on your group?

I don't think that's a thing as long as they are members of the group.

Blacklisting is the option you have to ensure members of your group can't join your giveaways.

I just want to be able to be the only person able to host a giveaway on my group's name.

It is not possible the way you want it

I mean you could make it a rule for your group to not give any games to the group and kick those who ignore the rule Maybe you should make a group discussion and only allow yourself to post there, and that can be the hub for your giveaways for the group :)

The best choice would be to simply ask the person doing the giveaway to not do it again. The person, with the best intention, probably thought it would be fair to give back to the group