I don't want to be discouraging, but a major publisher requested a full from me 2 1/2 years ago. I never heard from them again, even though I sent a follow-up letter one year after I sent the full. It's my understanding that a request for a full is not enough to give you an edge with an agent. I f you want to get a publisher interested first, then wait until they're ready to make you an offer, and then turn to a list of agents you've looked into and can query (preferably by e-mail). Tell them you have an offer.
I strongly advise trying to get an agent first, though, for the simple reason that there are fewer publishers than agents, and once they turn you down, those doors are shut.
I just received an offer from an agent, who already gave me great feedback on my novel once (I revised & resent) and now we've come to an agreement on a second round of revisions. I did not, of course, intentionally send out work that I knew wasn't ready. But the fact is that it wasn't quite ready. It doesn't matter how many agents rejected it; that won't affect its chances at getting published. If I'd gone to publishers first, that would be a different story. She'd be limited in where she could shop it.
It's tempting to go straight to publishers, especially when you get the requests for partials & fulls, even a nice, personalized rejection on an unsolicited full, as I got once, but based on my experience, it's not the best route. One of my novels (one I consider one of my most promising) was rejected by so many publishers, it will be years be for it can be shopped again, if ever.
My experience: four years querying publishers= nada
Nine months querying agents= representation, great advice on my writing, and hopefully a better chance at publication