Cassidy,
I would not say that しまう is mainly used for expressing regret at doing something. To me, it is more like, something is gone and we cannot bring it back.
In the example ケーキを全部食べちゃった, the cakes are completely gone, with しまう/ ちゃう/ちゃった expressing the idea that there are no more cakes.
For me, the sentence 今日は飲んじゃおう! is more like, "I am going to drink until I can't drink any more!" or perhaps, "I am going to go out and get really drunk, no matter what!" or "I have decided to go out and get really drunk, and I will not go back on my decision!"
Yes, there is a sense of regret in all of these examples, but I do not see this is the main meaning of しまう.