That is a strange one. Have you upgraded your headlights or the bulbs recently !
There is no headlight relay in the early Corollas. The headlight current passes through the contacts in the headlight switch on the column. I can only suggest that when the headlights are switched on, the +12V to the dash assembly drops off somewhat. The fuel guage is fed by a regulated voltage less than 12V, so that the guage always reads correctly. Are the headlights a bit dull also. I would measure the +12V at the steering column, with the headlights on & off, & see what the difference is. The regulator for the fuel guage can use a 7, 8, or even 10 volt regulator If the head lights are dropping the +12V supply to the dash, to less than 10 volts, that may be what is causing it.. My suggestion; is to have the headlight switch control an automotive relay only. Use the relay's contact to switch the +12V directly from the battery, via a 15A automotive fuse; to the headlights.
P.S. Have a read of recent thread, which may help you shed some light on the issue.
https://www.rollaclub.com/board/topic/85671-ke55-cluster-not-working-properly/#comment-738595
Cheers Banjo