Atwood GCH 10A-3E uses separate controls for Gas and Electric so you have 2 issues....one DC for Gas and one AC for Electric
Gas uses 12VDC from a Fused Source to the On/Off Switch
From Switch DC goes to Thermal Fuse (clear tubing) to/thru T-stat to Circuit Board via BROWN wire
T-stat closes for heating at 110*F....opens stopping heating at 140*F
From circuit board with t-stat closed DC goes to/thru ECO to Gas Valve
and Spark Electrode for ignition
So verify you have 12VDC to/from Switch and on BROWN wire
Could be blown Source Fuse, Bad 2a fuse on circuit board, blown Thermal Fuse (blows at 190*F) or T-stat not closed
Electric uses 120VAC from 15A Circuit Breaker to On/Off switch then to/thru T-stat then to/thru ECO to Element
OEM On/Off switch is on backside of Water Heater Tank on the protective cover over the set of t-stats and element
ECO is a Manual Reset if tripped.......small hole in cover-use pencil to push in the reset button on the ECO via the hole
IF you have an On/Off Switch inside the RV that is in addition (RV MFG) to the Atwood OEM switch and is wired in series with OEM---both have to be ON
Check 15A CB....verify you have 120VAC to/thru switch(es) , check if ECO has tripped/manual reset
Element could be burnt out......dry fire and they burn out
*Black & White wires removed...continuity between black & white terminals. No continuity...burnt out
Continuity test between White terminal and Element Flange.....continuity then element is shorted
Models prior to 2004.....6 gal/10 gal use same wiring diagram
Here is the 12VDC
Here is the 120VAC