I have an urge to pee and a full bladder feeling but it feels like it's on one side?

Get checked: A urinary tract infection can still cause these symptoms. See your doctor as soon as possible or go the the emergency room if pain is severe as it could also be a kidney stone

Here are some ...: Now that you experience significant urinary symptoms, you need to objectively confirm the clinical suspicion by further reviewing detailed Hx + proper physical exam + testing as needed such as urinalysis, residual urine, etc. so to deduce possible Dx and receive reasonable Rx timely. So, proceed with the plan above. Best wish ...