Water heater - propane problem

Had the exact same sequence issue with my central heat. Dirt dauber on the end of the burner screen. Fixed that and found my propane tanks, although low, did not have enough flow to stay lit.

My heater does not have an electrode that must be heated to open gas flow. Locks out after 3rd attempt. Gas flow is directed to igniter.

Your water heater using the SAME circuits/t-stat-ECO & thermal-fuse for electric and propane so if electric is working (heating/shutting down/reheating) then problem is NOT with any of the above.

Problem is with spark electrode, gas valve solenoids and/or burner assembly

Firing off, running for few seconds then making attempts to light off should have triggered 'lockout' and RED Fault light at On/Off panel.
That indicates spark electrode issue:
*dirty...carbon/soot build up....clean with emery cloth
*not being engulfed in main flame...tip red (needs to be in main flame to generate 'proving' signal to circuit board----no signal/circuit board shuts it down)
*incorrect spark gap.....S/B 1/8"
*burner tube not properly aligned.....gas MUST flow directly down center of burner tube in order to properly mix with air for proper combustion. If burner tube not aligned the propane will bounce down side walls of tube , not mix with air and not light off/stay lit
*flame spreader on end of burner tube not 'squared' (parallel) with end of tube...propane doesn't properly flare out and ignite/stay lit
*main orifice dirty.....unscrew from gas valve, soak in alcohol, blow out with air (no poking anything thru it)

Gas valve solenoids get 12V DC poser at same time spark electrode gets power,
You can FEEL them energize (only for 6-8 seconds unless main flame proves)
Make sure connections/terminals are clean/tight (RED--power/Black--Ground)

You can find the service manual at Chris Bryant's Site. It seems to be common for Atwood water heaters to have trouble with the edge connector to the logic board. Unplug, clean board contacts with a pencil eraser, inspect the contacts in the connector and re-spring loose ones with a dental pick. Might just do it.

"No clicking" could simply mean the thermostatic switch is NOT calling for the burner.

No clicking would indicate no power to circuit board.
Cerify continuity on thermostat and ECO.
Check for thermal fuse adjacent to the thermostat?eco. Normally hidden inside tubing. Should have continuity.
Check ground connection on the propane valve
Check continuity on each of the coils for the propane valve. Coils should read the same.