Welcome to the forum. I'm happy to answer this. I'm so sorry to hear you and your husband have been so turned into knots over this. It is unnecessary. You, your husband, and your doctor all have some serious misunderstandings about HPV. My advice to follow assumes what is stated or implied: neither you nor your husband has had other sex partners in recent years. Of course HPV is sexually transmitted, and most genital warts show up within a few months of being infected with HPV. But there are plenty of exceptions, and based on your story, I assume you are among them. (If my assumption is wrong -- and if so, I would suspect your husband of infidelity, not you -- then you both need to be tested for other STDs as well.) So here we go:
First: HPV infections can reactivate years later, which is the most likely scenario here. Either or both you probably acquired the infection causing his warts years ago, from other partners before you were married.
Second: You both are infected with the HPV strain causing his warts, or you had it in the past and your immune system cleared it. Either way, you can no longer be infected and cannot reinfect him.
Third: Warts typically become apparent a few months after catching HPV, so even if you didn't have previously, there is no point in changing your sexual practices at this time. Your doctor is dead wrong on this. Stopping exposure how, or using condoms, is equivalent to closing the barn door not only after the horse has escpeted, but after the horse is galloping over distant hills. It won't protect you at all, or him. I tell my patients in this situation to make no changes at all in their sexual practices.
Fourth: There is no need and no benefit from the HPV vaccine (Gardasil) in this situation. I don't understand "barely meet the deadline". The standard reccommendation for HPV immunization is that there is no need after age 26. I am unaware of any guideline that would either promote or decline immunization after age 44.
Fifth: There is no rational reason for your husband to have delayed treatment for his genital warts. I'm glad he has started treatment now.
Those comments address your closing questions, directly or indirectly. But to be sure there is no misunderstanding:
"Is it possible for genital warts to lay dormant for this long and for only one of us showing symptoms?" Yes, indeed. This is not at all rare.
"After receiving the gardasil shots will i be protected in the 90 % chance of it being HPV 6 or 11?" Probably not. As noted above, it's too late: if you weren't infected previously, you undoubtedly have caught his HPV infection by now. You can assume he had it for at least a few months before the warts showed up. Second, it is likely you either were infected all along or are already immune due to previous infection. If so, Gardasil won't help.
Where to go from here? First, discuss all this with your husband. Second, consider printing out this discussion as a framework for further discussion with your doctor. (Should she wish professional level clarification, I would be happy to exchange emails with her. It wouldn't be hard to find my email address with a little online searching.) Third, Gardasil is harmless, and now that you have started it, there is no strong reason not to continue, as long as your insurance covers it. But I don't expect it will benefit you. Fourth and most important, discuss all this frankly and sensitively with your husband. It seems he has been taken aback and perhaps frightened by these developments. But truly there is no need. This really shouldn't be the big deal that he has made of it.
I hope these comments have been helpful. Best wishes, good luck, and happy holiday.