Quiet professional seeking room near Caltrain (Palo Alto/Menlo Park/Redwood City/San Carlos/Belmont) — available early to mid August

I'm looking for a room to rent in a house, ideally within walking or biking distance of Caltrain in Palo Alto/Menlo Park/Redwood City/San Carlos/Belmont. I'm open to a room in a house, a room in an existing lease, or a shared living situation, as long as there's a private bathroom and it's within walking or biking distance of public transport.

A bit about me: I'm a graduate in the legal field, about to start a new job. I'm quiet, tidy, and keep pretty regular hours — mostly work, study, and the occasional hike on weekends. I'm hoping to move in around 13-14 August, and my budget is $1,200–$1,600/month.

If you know of anything or anyone renting out a room, I'd appreciate you passing this along.
Thanks