Is there such a thing as a course for acting and dancing WITHOUT singing?

If you want to audition for musical theatre, then yes, you will be expected to sing at audition. As stated above, traditionally places like Perfomers and Bird were more for dancers, but I think that is changing. I think you can be at Performers and not be a good singer, but you would still need to sing in the audition, and then if they like you enough for your dancing they may be prepared to deal with you not being a singer. Bird are definitely trying to go more triple-threat and Urdang seem to be turning away amazing dancers in favour of singers, although they have split into Urdang and The Dang and the latter is more for dance. Would you look at professional dance courses instead of MT?
Alternatively, why not audition for an Acting course where movement is important. They all contain movement to some extent but places like East 15 do a physical theatre course.
My last suggestion would be to look at some of the universities. It's not the same kind of training as drama schools, but it depends what you want the degree for. I know someone who went to Winchester Uni and did drama and dance who is now a secondary school teacher of... drama and dance. Would that sort of thing work, or do you want to perform yourself? If you want to be on stage, then I would decide whether you are an actor or a dancer first and foremost, and go that route.