Do you still have contacts with any of those individuals? If not, fix that and re-connect.
Personal connections are usually the best bet for this type of job. Look for the friend of a friend of a friend who knows somebody at a game studio, and become their new best friend. Talk to them, and figure out how to help them make a position for you an associate producer.
Don't wait around trying to win a lottery at a publicly-announced entry-level opening. You need to intelligently navigate your social network and actively knock down the barriers with tact and diplomacy, politely yet firmly getting people to see that they need you. It will be good practice, because that is most of what a producer does daily.
It's been a few years since I've talked to anyone at my internship, but I did have a few mentors, got along with everyone while I was there, and left on a good note. The company is so small I doubt they'd be able to just create an opening for me, but I bet they'd have some good advice or leads for me and if nothing else it would be good to catch up with some of the guys I used to hang out with. The college I work at has a game design program too. I don't want everyone to know I'm looking for a new job just yet, but I bet some of the teachers could probably introduce me to some people in the industry.
I'm very eager to get a job in the game industry as soon as possible, but at the same time I'm willing to wait for a position that's a good fit for me and with a good team. I'm going to aim for IT or an entry-level business position, but will stay open-minded. I think for now I'll start working on networking, keep applying for public listings, keep studying the industry, and stay alert for any opportunities.
Thanks for the advice!