How do you deal with someone you love that has a fearful avoidant personality and run hot and cold?

As someone with it: patience and clarity. Make it as obvious as possible that you mean them no harm, go easy on them - as in don't overly criticize everything they do thinking it will push them - they are the exception and you will hurt them in the process, and simply wait. None of us CHOOSE to be like this; in my case specifically, I have never had anyone in my life I could actually trust and was therefore convinced that everyone was out to get me.