How should the ground wires be connected when changing my dryer's power cord from 3 prong to 4?
The installation instructions show you have wired it properly for a four wire cord. I would check the receptacle to verify it is wired correctly. 

Before you do anything if you haven't done it yet download the installation instructions from Maytag. It shows the green wire being connected to the center screw is just a placeholder for a jumper that is connected to the top of the neutral terminal via a spade stake-on. If you connect it to the ground you will be bonding the neutral to the ground as required for a 3-wire NEMA 10 connection. 

If after connecting cord correctly I would confirm that the four wire receptacle is being fed by four wires, specifically to confirm a white wire is connected to the L-shaped terminal, green to the opposite terminal, and 240v between the parallel flat blades.

Beyond that your problem sounds like it could be internal to the dryer, way beyond my ability to advise without more pictures.

If you move the jumper and the dryer receptacle is missing a functioning neutral the dryer (if in working order) would illegally operate by running the 120v circuitry on the ground.