[Newbie2]

I have a White-Rodgers thermostat, Model 1F82-162, that was installed in 2003 or 2004. It runs my outdoor heat pump and is also attached to an indoor blower (furnace?) unit in a closet. The heat pump is the same age as the thermostat, the indoor unit dates to 1984 when the house was built. Until last night, everything seemed to be working fine with the system.

Last night I noticed the backlight on the thermostat was not on. That was my first clue that something is wrong. Next thing I noticed is that the heat did not come on when programmed to. Neither the heat pump nor the blower was going. So I started checking and testing:
Flame icon was flashing on the screen, indicating it was trying to heat.
Day and time were wrong (said 12 on Mon when it should have been 9 pm on Wed). I reset the day and time. It stayed in memory for some time, but eventually it reverted to 12 on Mon.
I set the switch to Emergency heat and the display responded, but nothing turned on.
I replaced the 2 AA batteries inside the cover. Reset the day and time again, checked that the programming had stayed (it did) and turned it to heat, but nothing turned on.
I went outside and flipped the circuits for the heat pump and furnace off and back on again.
I looked at the outdoor fuse for the heat pump, but don't know enough about it to tell if anything is wrong. It's all behind a plastic cover which I did not remove.
I pressed the Reset button on the inside of the thermostat and waited for any change, but heat never came on, although the thermostat itself seemed to reset (the current temperature showed up as 71 and gradually went down to the actual temp of 57, the day and time had to be reset again).
I checked all the configuration items on the thermostat and they were set at factory defaults.
I set the desired temperature above the current temperature, and waited to hear a click like it says in the manual, but I never heard it click, although the flame icon would display and eventually start to flash.

What part of my heating system might be faulty? I'm thinking I need to replace the thermostat, but don't want to take a stab in the dark. I'm good at following instructions and thinking logically, but am not an electical or HVAC expert.

Thanks for any input!