Question
QUESTION: Is solenoid bad? Water flows if bleed screw backs off. Water flows if solenoid is backed off. Replaced diaphragm. I can see solenoid tip pull back after I removed it and activated switch. Flow control backed off, not closing system.

ANSWER: Thanks for you question.
It appears the valve won't close after an electrical activation. If the plunger inside the solenoid is moving when you activate the switch, the solenoid is OK. It's action of moving back into the solenoid is what make the valve open. When it drops (electric off) it seals the tiny hole in the center of the valve body. When you push the plunger back into the solenoid it should spring back freely. There is a spring behind the plunger. Try removing the plunger and checking the spring. This area must be perfectly clean.

You also mentioned the flow control is backed-off all the way. Try turning it in a few turns or until you hear noise from the valve. The valve need a few pounds friction loss for it to operate correctly. By having the flow control in the full open position the diaphragm needs to travel a long distance to close the valve. Shorting that distance may help.
If you continue having problems. I'll need the size of the valve ie; 1" or 1-1/2" and also an approximate water pressure. I may be able help further.

---------- FOLLOW-UP ----------

QUESTION: Thank you for your reply. Plunger in the solenoid springs back when I push it with my finger. Solenoid pulls in when I activate it, looking at it out of the valve body. Water flows freely if I back out the solenoid, so hole in cover is open. Trying different positions with flow control does not help. Its a 1" Ridell.

Answer
Thanks for giving me more info.
The only possibility I can see at this point is the exhaust port under the solenoid is plugged.  When the solenoid is activated, it lifts the plunger and releases the water which is holding the diaphragm closed. That's the water you see squirting out when the solenoid is removed. When operating correctly, that water dumps into a small hole under the solenoid. It is usually to the side of the center hole toward the end of the valve. If that hole is plugged it would stop the valve from opening. It is possible the clean the port, but it may prove more work than just replacing the valve. You will probably need a magnifying glass to see the port and to try to clean it.
I hope this helps.