Re: yeast infection or UTI, maybe?
Burning while urinating and feeling like you have to pee all the time are definite signs of a UTI, and UTI's are common for some women after sex.
Regarding the itching, if you had a yeast infection you'd likely also have itching around the vulva area, but this doesn't mean it's not a yeast infection. Deep vaginal itching isn't a sign of a UTI though, so you might have more than one issue going on. Really hard to say what's causing the vaginal itching.
You really need to be seen by a doctor for this to figure out what's going on, and the sooner the better. Be sure to mention/write down all of your symptoms. If you start to run a fever, have mid-back pain, or urinate blood, be sure to go to urgent care/ER ASAP as those can be signs of a kidney infection that can happen sometimes if you don't treat a UTI soon enough.