Could a saturated drain field cause a toilet to back up?
As to backups into the sewage system, yes, a saturated drain field could cause that, no place for the grey-water to go.

Check around the inspection hole on the septic tank, should be a lot of leakage if that's the case as the tank will be full with no airspace if it has clogged outflow.

Excessive rain and drowned drainfields is a common wintertime issue here in Western Oregon. It's one reason our local city boundaries got massively redrawn to include a large area of the valley into the sewer district as the high water table and precipitation meant surface water was a health hazard.