I just received Dysport for the first time between my eyes and my eye always feels like something is in it, is this normal?

in all probability the eye discharge and discomfort dont have much to do with injections into your glabella (between your eyebrows) - injections in to the crows feet has been shown to worsen dry eye symptoms in a small proportion of people, however. 
there are several questions here - is the eye red, irritated, itchy? perhaps this is related to an allergic conjunctivitis? usually this is bilateral, but unilateral symptoms are possible. you should probably be seen by an ophthalmologist, and not simply your injector (unless he/she happens to be an ophthalmologist) 
good luck

This is obviously not a normal problem and needs evaluation from the doctor. I suspect that this is unrelated to the injection. It sounds like a possible injection or even a foreign body in the eye. Please get evaluated.