I agree with erg923.
Edit: After writing all below, I just read that this cannot be expunged. If so, if this were me (based on limited info here), I would start the executive clemency process, (seriously, can you work with an attorney to contact your governor's office?....maybe not the president, of course) regardless of the 'no timeline.' (One may think...'no one has time for that,' but why not, if you're a superstar now? And I say this with sincerity mixed with some naivete of the legal system). Doctoral programs take 5+years, you could have what you need before time to apply for internship (where you will be asked to check a 'ever convicted of a felony'? box).
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Search an old thread about the getting the conviction expunged. I don't know if you can do this, but you should contact an attorney about this (if this was a first-time offense, and you can prove to the courts that you are a 'different' person with many improvements, they may be able to seal this conviction from your record; also discuss with the attorney how, or if, you report on it, once it is sealed/removed). If you can't afford an attorney, see if you can find one
Pro Bono or ask a court advocate how they could help you seek legal help.
You sound like you've done so much academically, and I am from the school of thought that we can improve from past transgressions (like how old you were in 2006 may say much compared to now).
If this were me, I would put some effort into looking into to expungement first, then contacting state licensing boards, then figuring out personal statement (only b/c some programs will see the felony and decide to reject - so I wouldn't waste time there first). And if the felony cannot be removed, I would contact the universities' graduate school offices - not the individual psychology depts - (maybe anonymously, at first) to see how felonies fare for admission there...then, if some programs are foregiving, I would most certainly address in 1-2 lines in the personal statement (maybe even vauguely, something like "I've improved from past transgression in these ways...." (i.e., then, when on an interview, they'll ask what your transgressions were and you open the dialogue to talk about how you've grown constructively).
Good luck!