Camping-mama --
Can you clarify your 'zero' temperature comment -- Fahrenheit or Celsius? Reason for asking is the ambient temperature "sets" the propane vapor pressure from your propane tank. At 0 F, the vapor pressure is ~35 psi which should be very adequate for your propane stove and furnace which need 11-12" water column (~0.5 psi) of propane pressure to operate.
Based the description of your troubles, a potential problem is the pressure regulator for your propane tank is sticking closed. A couple of gentle whacks with a hammer on the regulator may free it up. Replacing the regulator is a relatively inexpensive, simple DIY job.
Presume you have a propane regulator for two tanks that looks something like this one (?) -->
https://www.etrailer.com/Propane/Camco/CAM59005.html
Also, if your propane tanks are less than ~30% full, in cold weather there may not be enough propane to maintain an adequate supply pressure as the propane vaporizes coming out of the tank. For cold weather use, it's best to keep propane tanks at least 50% full. This is a good article on propane tanks / propane use in cold weather -->
https://upgradedhome.com/can-propane-freeze/
Let the forum know what you learn about your propane system.