I just saw this posting and thought I might throw my ideas in though its a bit late. I am not sure how set you are on Linguistics programs per se but will offer suggestions in and out of Linguistics (sign language acquisition in deaf children, though not my current research right now, is pretty much the field I work in -- mini disclosure I'm a student at one of these programs)
Diane Lilo-Martin (in Linguistics) and Marie Coppola (Developmental Psychology) do work on language development and American Sign Language, you could look into those departments. Coppola also works on Nicaraguan Sign Language. If by cognitive linguistics you mean "not generative", both professors have a strong generative streak so you may want to be wary.
Rachel Mayberry in Linguistics at UCSD also has done work on language development in Deaf children, I think she has also done work on literacy and deaf children. UCSD is also famous for sign language research and have several professors across Linguistics, Communication Studies, and Psychology who do research on sign language.
At Gallaudet you have a few options you might find interesting -- I think Guarav Mathur in Linguistics works on language development, and the students in PEN program (Educational Neuroscience) often do too. Gallaudet linguistics program is also pretty cognitive (anti-generative) but not the best with funding. As for ASL, yes you'd want to be as good as possible.... the requirements vary by department and you don't have to be fluent but I think you have to be able take classes in ASL for Linguistics at least.
David Quinto-Pozos at UT Austin is a sign language linguist who has recently shifted to more developmental (and clinically-applicable) work, and I think is theoretically oriented towards cognitive linguistics. Richard Meier I believe is also still there -- he works on ASL but not development.
At U Chicago, Susan Goldin-Meadow (Developmental Psychology) has done work on language creation with deaf children (homesign) and now collaborates some with Diane Brentari (Linguistics), who is a sign language phonologist, who I think has done a bit of L1 ASL acquisition (not much) -- but between them you could probably do work on language acquisition and ASL.
If you are interested in psycholinguistics and sign language both David Corina at UC Davis and Karen Emmorey at San Diego State do relevant work (just not with kids) -- oh and Jill Morford at University of New Mexico does work on bilingual language processing with Deaf and hearing ASL signers
I think DCAL, a center at UCL in London for sign language research has done developmental research, but don't know much about who, or if it still running. Also its another country and another sign language (BSL), so there's that.
P.S. if you were interested in also the possibility of just an MA first, I think Gallaudet and New Mexico at least have a fair number of students who do an MA there and then a PhD elsewhere.
