Keep my head above water is the normally accepted comment. We (English Speakers) do not use the article the in normal conversation. By the way, to keep ones head above water is usually meant to have financial success or to survive financially.

Keep head above water = an idiom that means stay ahead of or on top of of a difficult situation. Example: I am so busy at work I can barely keep my head above water. = I can barely get my work done. In the US, "keep head above water" does not necessarily refer to financial matters as my sample sentence illustrates. In non-idomatic usage, include the definite article. Example: The swimmer kept her head above the water.

Yes, you are correct. To keep your head above water is an idiom which means to stay out of difficulty, financial troubles etc. With the article "the", it means literally "the water" as in your example of drowning.