What’s going on with the running toilet

Knowing where the water lands gets you in the area, but what you really need to know is where the water came from. Is the tank cracked, is the supply line failing, is there a bad connection on either end of the supply line, or is the valve itself going.

If the tank is cracked, you'll have to buy a new one, and it's a large project if you're not used to working on basic plumbing jobs or are skirmish around an open toilet.

A bad supply line can be easily replaced and it's even easier if it's just the washers.

A bad valve may just have some dirt or buildup in it that will be fixed by opening and closing it once or twice with water running through it (an empty toilet tank). Though keep in mind any dirt may then get in the toilet valve and could cause it to run slowly until you clean it out. If that were to happen, just post another question. It could also be the packing material in the valve, which I've never touched on this type of valve. If that's the case, you may be able to minimize it by opening the valve all the way. If the valve is too far gone, you'll likely need a blow torch and solder to replace it. This can be a DIY job, but if you get in over your head, the water for the whole house will need to be shutoff until you can get a plumber out.

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.