At the other end of the pedal itself, there is usually a screw or a bolt attaching it to a horizontal wooden stick. By turning that screw or bolt you can adjust the contact between the pedal and the dampers. This is a temporary solution only; a properly working piano needs the dampers to respond exactly the same. First, perhaps, check that all the wooden sticks are connected as they should! Remove the front plate where your legs are!
My advice is "Stay away from the inside of the piano unless you're a piano technician". Would you play around with your car engine or gear box if you find that something is not right?
It's been my experience that a fully pressed pedal has no dampers touching strings. However I find on pianos with shallow pedalling, the final position of the dampers is closer to the strings than on pianos with deep pedalling.