It doesn't sound like a UTI (although you should report any urinary symptoms to your health care provider, as UTIs are a common cause of preterm labour). It could be that your uterus is pushing down on things but you shouldn't need to strain and in fact you could make the problem worse by straining on your urinary sphincter. I would strongly recommend speaking to your health care provider about this - many hospitals have clinics attached to them especially for dealing with urinary issues, and while you aren't incontinent, having trouble communicating your wishes to your bladder might need some help from someone experienced with these issues!

I sometimes have trouble getting started and I find that just gently rocking backwards and forwards while I'm sitting down, and sometimes even physically lifting my uterus up and off my bladder!