A lot of the newer car TPMS systems are using the ABS system sensors to determine outer tire circumference, thus a damaged tire or a tire with extremely worn tread can set off the TPMS light (IE driving on the donut spare). If you have this ABS style TPMS system, it is imperative that all 4 of the tires match closely in outer tire circumference. If you tagged a curb really hard and messed up the tire, it can be the reason you keep getting the TPMS warning light.
ETA; You should buy a $5 tire pressure gauge for your glove box before trusting a tire shop wanting to sell you a new set of tires.
As others have said, it a simple dunk the tire in water to find the leak. It could be a valve stem, tire bead, or cracked ALM wheel.
Have someone pull the wheel off the car and take it to a bathtub where you can dunk it and spin it under water to find the leak.