Years ago with the HAI Omnistat RC-80 I had many issues with the power "stealing" piece of it and my furnace motherboard. An additional piece that was causing issues was the HA to HAI panel talking to the thermostat (pegging the thermostat at 90 for no apparent reason). First piece initially was disconnecting the HA box from the HAI panel (eventually downgrading HA HAI plugin software). Very low WAF with this issue.
I have a friend whom has an HVAC company and we kind of went back and forth with the issues. He proved to me that it wasn't the furnace MB one day by removing the RC-80 and installing an old fashioned very manual Honeywell thermostat (the old circular type). I was actually to a point of monitoring the LED status lamps on the motherboard (blinks and whatever) and had left a VOM hooked up to watch voltages. Some of the testing also involved using jumpers on the furnace motherboard to manually do what the thermostat automatically did to validate the furnace motherboards functions. My HVAC friend questioned my "need" to automate my thermostat via a computer and convinced my wife that it was not necessary.
Initially had tried all of the "fixes" recommended by HAI (comm wire, thermostat power supply module isolation box (30A00-2), etc) and finally what fixed it was a separate transformer for the RC-80 with a relay isolation 29A00-1. Initially even that caused some issues. I think what had happened though was that the on and off and constantly making changes had changed some of the initial base settings on the thermostat. I eventually went to "readjusting" every default value manually (without connectivity to the HAI panel) on the thermostat and it worked.
So in summary I would kind of use the same approach that my HVAC friend did for helping me with my HVAC/RC80 problems.
1 - disconnect the Omnistat from the furnace motherboard
2 - validate function of the furnace motherboard voltages and whatever (if the MB goes out it may be more expensive than an Omnistat to purchase).
3 - connect a "manual" thermostat to see if it functions fine with the wiring and furnace motherboard.
This piece will validate function of your furnace motherboard / manual thermostat.
I would then try installing a 29A00-1 with its own transformer to see if that helps you. I've left mine in place from when I had my RC-80 so I don't really know if the new RC-1000 would have caused some issues with my furnace. The RC-1000 has been in place now about a year or so with no issues.