Emma,
You need your residency to legally work in the DR. That being said, you can probably get away without it and not get bothered unless it is for something else. I know lots of people that work there for years without the residency.
I got mine through an attorney. I couldnt have done it without one. The moderator of this board Fabio Guzman has a good reputation and you may want to find one of his posts and click on the Email button under his post and inquire with him. I am sure that you can come to some sort of arrangement to get it done if that is the course you take.
Look up Bushbaby here on this board also. He is a Brit that has been in Puerto Plata for 10 years and he is a nice helpful person. I am sure he would bend over backwards to help another Brit out. Hell all you have to do is bring him pipe cleaners or some other stuff that is hard for him to get and he will help you plenty
Takes about 3-5 months to get the residency if I remember correctly. Cost about a thousand dollars US also just so you know.