I write fiction in addition to doing math and pride myself on my abilities in both media. Both my proofs and my stories are verbose and *highly* detailed. This often frustrates people who know as much as I do on a topic or more, because they view me as being too pedantic. However, novices appreciate the extra detail, and that is all that matters to me. Accessibility is always a priority of mine.
Anyhow, as for YOU, some of the tips I have to give are:
1) Always write the statement(s) of the question/exercise/result/problems you are being asked to prove BEFORE you write out a proof.
2) Doing a proof is actually two different tasks. The first task is for you to find A proof; your second task is to WRITE UP your proof. To that end, do not bother trying to write things up neatly until you have solutions to ALL of your problems (or however many you can complete before the deadline). Once you have your solutions and are confident they are correct, take your time to focus on writing up your solutions neatly and carefully. It is MUCH easier to focus on improving your proof presentation when you only need to worry about how you write up the math.
3) Clearly indicate everywhere and anywhere where you utilize the hypotheses given in the problem.
4) When I write up proofs with computations, I like giving justifications/explanations in parenthesis to make them easier to follow.