It is hard to know exactly how to answer your question without having all the specifics of your situation. However, the basic problem is that unemployment and disability are for the most part inconsistent with each other. For unemployment, you typically need to be available for work, capable of working, and seeking work. For disability, you are saying that you are not capable of full-time employment.
Social Security may hold it against you if you are collecting unemployment while trying to get disability because of this apparent discrepancy. However, there are circumstances where there actually is no inconsistency between the two programs. For example, a person who is 50+ years old and limited to sedentary work could be found disabled by Social Security's rules if they haven't done sedentary work in the previous 15 years. If such a person is collecting unemployment while looking for a sedentary job, then there is no inconsistency.
Your best bet would be to contact a local attorney who specializes in Social Security disability. Many attorneys will offer a free consultation to see if the facts of your case warrant a Social Security application at this time or if it would be better to hold off for a while.
This is a difficult question to answer -- by applying for unemployment you are specifically stating that you can work. If you apply for Social Security benefits, you are stating that you cannot work.
However, you have to eat and put a roof over your head. Please consult a local social security attorney. They will ask you questions about why you cannot work and your disability. Perhaps you meet a "listing" and would be successful on the initial application.