I have an Associates Degree and and currently in an NLO class with 8 people, 2 of us Nuclear Power Associates degrees, 2 Navy and 4 Bach. degrees (various engineering). My suggestion would be to apply, apply, apply. Many house positions these days are diverse and do not just take Navy nukes anymore. Those programs are out there for a reason and many companies like them and the people they produce. I am in a position that only took me 2 years to get to, in respect to the 6 years in the navy and 4 years at an expensive university. I think you have a better chance than you think! Good luck.
Yes, keep applying for work.
The associates is plenty of education to show the plants you are:
1. trainable.
2. willing to finish something you start.
That's all they need to see...if they need people. Right now you are fighting a situation in nuclear power where supply exceeds demand (due to a bazillion schools pumping out people with associates degrees), so it's tougher to get in, no matter what your quals are. Maybe it's easier to get hired on the non-nuke side? I would apply everywhere that met my minimum requirements (location, pay, etc). That being said, don't just wallpaper the world with a generic resume - take time to craft each resume and cover letter to match the job and company you are applying with.
Further down the road, you will probably want the bachelors degree if you want to climb the ladder. If you can't find a job, you could start on that now. BUT, as Birch said, the associates is sufficient to get you in on the ground level.
Good luck.
Full disclosure: I graduated with an associates at another point when the industry was reaching 'flood stage'. The school I attended closed their nuclear programs a year later, because they were having no success placing graduates. I couldn't get hired for over six months, but just kept plugging away with the applications. Eventually, I got my foot in the door and...nuclear power was VERY, VERY good to me!