Smart Trashbin metal detector with coin reward... need help

As a first step you should test the pure function of the servos by using a sketch that is well known for working. By using such a sketch you can check if the hardware is really OK.

What is the real 5V regulator that you are using?
7805? LM317? something else?

TO92 housing?
or
TO220-housing?
or
something different?

Post the exact type of your 5V voltage-regulators or post a real picture of the voltage-regulators taken with your smartphone

What kind of proximity sensor is this?

Post the datasheet of your proximity sensor