In case of grade crossings of state highways over such railroads, the Department of Transportation, after due notice to the railroad, corporation, or operator, and hearing the railroad or operator involved, if application is made for such a hearing within ten days after receipt of the notice and after finding that the public safety, convenience, or necessity require it, shall have the power to specify the character of the grade crossings, and the operator of the railroad shall construct and maintain the crossings to meet the specifications of the Department of Transportation; provided, however, that the power to specify the character of grade crossings granted in this section shall not extend beyond five feet on either side of the center line of the track; provided, further, that the Department of Transportation shall have the power, in matters relating to such grade crossings, now pending or hereafter arising, to enter into such agreements with operators of railroads pertaining to the construction thereof as in its judgment may be to the best interest of the State, and to agree to pave the area across the tracks after the area is otherwise prepared for paving by the operator of the railroad. The Department of Transportation, with the advice and consent of the Attorney General, may waive any and all claims for penalties now existing, upon entry into such agreements.