The hard part is that the best need-based financial aid usually comes from the most selective colleges, some of which admit few transfer students. Merit scholarships tend to be less common for transfer students.
I take it UNH is not affordable? List price is about $31k if you live on campus, of which about $18k is in-state tuition and books. If you can commute there while living with parents, it may be barely affordable at list price after adding in live-at-home and commuting costs to the $18k of tuition and books costs if you would not get any financial aid there. But if you are not in commuting range and would not get any financial aid there, it is significantly over your $20k limit.
Others may be able to help you better if you mention what you dislike about your current college, so that they do not name colleges that share the characteristics that you dislike.