I think "did" work as a normal verb in this situation, but I'm not sure about it. I would like to have your opinion. Thank you!
To do one's best is an idiom which means to do something as well as one can, so the sentence may also read:
I did everything I could.
I did my best.
In both examples the verb to do has its literal meaning of to carry out an action/actions, so did, in the examples, is just the past form of the verb.