Should I mention my willingness to pay my own expenses in my cover letter?
No. Don't bring this up. They don't care, and those who will do the initial selection of resumes aren't interested in that at all.
In many cases, they won't care where you are actually living (unless you have specific jobs where you are required to be in the office within X minutes during "on call" time, but that should be clear from the job description). It's just that they won't do relocation. But whether you solve that problem by paying for relocation yourself, or having a 3 hour commute should not matter.
It does mean though that you shouldn't bring up relocation fees during the hiring process. Bringing that up will just make them say "which part of 'no relocation fees' did you fail to understand" (although probably phrased a bit more friendly).
Just apply. The worst that will happen is that they won't hire you. Which they certainly won't do if you don't apply.