I think, before you do anything like finding another home for the second dog, you need to send your ex a letter IN WRITING, to be signed for, saying if you don't hear from him by (date), you will presume he has officially abandoned both, and will be proceeding to rehome the one you don't want to keep and keeping the other, bearing in mind that if these two are attached and have been living together for some time now it may be very hard on either of them to be parted. This isn't only about you much as I understand you may not have the finances available to keep 2 dogs.
Adopting legally (the one you keep) is difficult as posession is 9/10ths of the law but a dog is property and could, in law, be regarded as his dogs unless he replies in writing stating his intentions. Again you need to force the issue here by giving him a time-frame in which to act, beyond which these dogs are being regarded as abandoned.
I suppose if you want 'legal adoption' you could hand both into your local Shelter as abandoned, and then adopt one, with all the paperwork. I do think you need to take legal advice - I'd be talking to your local Shelter/Animal Control.