Had cold for week. now back where nose meets throat very sore hurts to swallow. feel mucus stuck there and coughs out bloody. do i need antibiotic?

Not likley!: Your symptoms are consistent with tonsillitis in the usual progression of most upper respiratory tract infections that are invariably viral! if you have previously had strep throat infections a screening swab &culture should be done, but unless you are among a community that carries that infection, it would be improbable for you to have strep. Rest, fluids, salt h2o gargles &lozenges can help!