I have a lesser, similar problem: my eyes don't track together, so one of them tends to "wander off" on its own and it looks like I'm not looking at the person I'm speaking with. Over time I've learned to fake eye contact; if I actually look directly at somebody it looks like I'm not.
I've taken two further approaches to this problem:
I participate actively in the interview. I sit up straight, have done my homework in advance, ask questions, and don't give dead-end answers. On that last: if they ask "Have you worked with such-and-such technology?", don't say "yes" -- say "yes, with (specific flavor) at (company), to do (task)". Don't over-fill; maybe they only wanted a yes/no and don't want you to talk for the next two minutes. But say something that offers a clear invitation -- you're prepared to discuss this more. The combination of appropriate posture and showing genuine engagement in the interview can make up for a lack of eye contact.
If I'm getting a feeling that people are perceiving me as disinterested because of the eye-contact problem, I'll say something to one interviewer, ideally a prospective peer or the hiring manager. I do this during the wrap-up at the end of an interview segment, after I've asked some questions. I say something like this: "You may have noticed that I have trouble maintaining eye contact. That has nothing to do with you or my interest in this position; it's a minor medical problem. I am very interested in working with you." Telling one person is sufficient; the interviewers are going to talk with each other afterwards, most likely.
With what you've described you don't have the "fake it" option that I do, but the others should help you.
As best I can tell, this has been working fine for me. At companies where I knew somebody and got informal feedback nobody's raised it, and I haven't had any "huh, wonder why they didn't pick me when there's obviously a good match" moments after in-person interviews. Mostly I do #1; last time I also did #2 I got an offer anyway.