I exist.
It's not a lot different from, "I am I, Don Quixote."
"Don Quixote" is an appositive, which means it can be eliminiated without changing the essential meaning of the sentence.
Therefore, "I am I," is a sentence.
Cogito, ergo sum. Here too, you can eliminate either one of the independent clauses.
"I think," is a sentence. "I am" is a sentence.
If "I think, therefore I am," is a sentence, so is "I think, and I am."
It's a little more difficult to justify "I do" on these grounds, but still possible.
Clearly, "to be" means "to exist," apart from it's use in expressions like "I am dumb."
But does "do" have a meaning in its own right?
Consider the adage, "Those who can't do, teach."
So I may say, "I can do."
Therefore, I can say, "I do."