Do modern aircraft have sensors to determine weight of the aircraft?
No, the 737 does not have such sensors installed because it is less expense to compute the zero-fuel weight (and thus takeoff and enroute and landing weights, by adding fuel on board) by taking the known aircraft empty weight and adding to that the weights of the people and things that are on board.

I've never heard of any attempt to install any such sensors on a 737. Some FMC software can use in-flight performance data to estimate gross weight and alert the pilots if that estimate varies too much from the computed weight, but that is an entirely different approach.