10 month old's cold/nasal congestion not getting better after 3 weeks, Please advise.?

I would give her the antibiotics, if she is having trouble breathing this cannot be comfortable or safe for her even if they don't work fully they may make it better for her. I'm sure you are trying to be a good parent by doing this but her body will still build up and immune system for it. I think it would make her more comfortable and much better, after all thats what they are for :) However it is totally up to you, you should be the one to decide! Hope she gets well soon :)