The 'year' rule has to do with how long someone has to GIVE you a gift, not how long it takes you to thank them for it!
I would send as many thank you notes as you can immediately, and then find someone to get that list for you so you can send the rest. Those notes should have gone out a week after the shower, not a year!
Also - no need to acknowledge the lateness of the notes. It will just sound like excuses - and really, everyone is busy with something. (You're not the only one moving/having kids/in school/etc) Just send a warm and genuine note for every gift, and you'll have done the right thing.
I can understand why people WANT to believe that they have a year to write thank you notes. But why anyone WOULD believe it is harder to understand. What would be the justification for waiting so long to thank someone for their generosity and in many cases even to let them know that the gift was received? No one (actually two, for a wedding) is THAT busy -- a whole year. You get a little more leeway for occasions like weddings when there are so many to write, but the basic principle is the same: you thank people as promptly as you possibly can for any kind of gift.
But if you can't be prompt, do it anyway! As you commendably are going to do.
And now you also know how important it is to WRITE DOWN who gave you what gift. First of all, it makes it possible to thank the right person for the right gift. A thank you note for "the generous gift" is nothing more than a handwritten form letter and makes it obvious the writer doesn't connect the givers to their gifts, but is just writing the notes as a chore (as Miss Manners once put it, "[t]he proper expression of thanks must be accompanied by a vivid detail, as convincing evidence that the reply is not a form letter but was inspired by the actual and particular present. ") It also makes it much nicer to use the gifts when you can think of the givers as you use them. After 31 years of marriage, I still do, and it gives me great pleasure.
Don't say anything about your lateness in your notes. Just thank the people as quickly as you can now. I might even go ahead and thank them for both the shower gift and the wedding gift (and anything else they did for you) in the same letter. True, it won't look so great, but honestly two notes close in time at this point will look pretty much the same. Think about each person and decide if this will work for them; for some it will. Like your favorite aunt, but maybe not for your mom's boss.
You're going to feel funny as you do it, but you'll feel great when you finish! And after all, it's about their feelings, not yours.
Good for you for not letting your embarrassment stop you. I remember someone once asking Miss Manners "When is it too late to write a thank you note?" and her answer was, "When the person who was generous to you is dead, and you have to live with the knowledge of your ingratitude."