Dissertation fellowships may target a particular topic, a particular group of recipients, or both. It’s a good idea to become familiar with fellowships which may be applicable to you given your topic early, even well in advance of when you would become eligible to apply. Make sure you meet the criteria of a fellowship before you put the time into applying.
When requesting letters of recommendation, you should ask potential writers at least one month in advance; provide all needed information if they agree to write, and follow up diligently to ensure that they have what they need and get a reminder before the deadline. Like you, potential letter writers have a lot of stuff coming at them all the time, and you have a great incentive to politely help them remember to follow through.