The buzzer won't stop buzzing unless you open the door or turn on the timer and the dryer will start just by selecting a cycle and closing the door.
From the description you give the most likely cause of both problems is the push to start switch. The motor switch is another possibility but I'd suggest you start at the push to start switch.
The wiring diagram for your dryer is in the 1st image below and it may be helpful to refer to it to diagnose this problem.
The buzzer is suppose to only buzz until the motor shuts off and the motor switch opens which breaks the neutral to the buzzer. When you open the door you do the same thing. It's possible the motor switch is stuck.
A bad push to start switch will also cause both problems. The switch can be checked with a volt/ohm meter if you have access to one and are comfortable using it. The switch is normally open. See the wiring diagram for a picture of this. When you push the switch the motor starts but it sounds like the switch is already closed.
If you have a meter check continuity across the switch. It should show an open circuit till you push the start button and then it should show a closed circuit until the start button is released and the circuit should show open again.
You can remove the rear panel of the console to access the switch. Remember to unplug the dryer first.
If you don't have a meter there is another way to check the switch if caution is used. One trick is to remove the white/red and black wire off the push to start switch. Let them hang loose but not touching anything. Try with caution plug the dryer in and select a cycle. Close the door and see if the dryer starts. If it runs then most likely the motor switch is bad. If it doesn't run then most likely the push to start switch is bad.
If you still want to check things then the other trick is to start the dryer with the push to start switch connected then after the dryer starts remove the black wire from the push to start switch if individually connected or both wires if on a connector. The dryer should continue to run and at the end the buzzer should only stay on as long as the motor is turning. Use extreme caution if you try this.
The push to start switch is part number 3395382 and can be obtained from the following site:
for about $11 dollars.
I hope this information helps to resolve this problem but submit another question with any additional details for further assistance.