Appropriate swimming attire for lap swimming / getting over anxiety

I recently got a membership at my local Y to start swimming laps for fitness as I’ve wanted to do swimming for exercise for a very long time. I tried swimming at the Y several years ago but felt intimidated by the lean swimmers who looked like they knew what they were doing and swimming super fast.

Within the last few years, I’ve worn swim briefs under my trunks for better support and comfort (and in case my trunks decide to come off during a dive then I wouldn’t be fully exposing my birthday suit).

I’ve come to the realization how comfortable briefs are and I wish I had the confidence to wear them when swimming laps, however I am a bit overweight but not morbidly obese and am self conscious about how others would perceive my appearance. Most people I see swimming wearing briefs are lean and not chubby in any way which would make me feel out of place or uncomfortable. I don’t want to wear trunks to swim laps either as I feel they would impede my movement and slow me down.

Does anyone have any insights on what someone in my situation should consider? I’m completely new to swimming for exercise and would like some advice to be pointed in the right direction. Thanks in advance!