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Honestly, with credit that poor, you should really consider waiting a few years and focusing 100% on the both of you improving your credit.
If his mid score is a 580, that's very poor. I understand you may want a house now, but you're very likely to get an awful rate with scores that low.
Not trying to be negative, but taking on the huge responsibility of a mortgage while you have outstanding collections is not a good idea.
My advice? Stop carrying a balance on all your cards entirely. It's both money thrown away, and is just hurting you because it's THAT MUCH LESS that would be going towards mortgage payments. I would be pushing for PFDs on those accounts. I doubt the credit union will PFD without a paid in full offer, but it's worth a shot. The Trident account you can probably get early-excluded in 1.5 years-ish if it's from 2012.
Really though, I would not obsess over a mortgage until all your debts are paid off and you've built up his credit, and looked at getting yours under control (since you imply it is not in good shape).