It's less than good practice to drain gutters into a French drain. The downspouts should be connected to a "tight line" system - 4" solid pcv drain pipe that is diverted to as far away from the house as possible. Maybe a retention pit or an infiltration trench or a stormwater pond. Water from your roof should be allowed to percolate back into the local watertable. It's just that you want to keep it out of your basement.

You could try and find where your roof drains ultimately terminate... if there is even a tight line system, so yes start where the downspouts go into the ground. To find the outflow look to the low point of your property. Look for a pipe sticking out of the ground in low areas of your property, if you're near a road side ditch the drain lines could drain there although many jurisdictions prohibit this practice.

If you're concerned that there is no drain system beyond the downspouts dumping water right against your foundation, turn on YouTube and find a how to video.

The ground can be saturated without being squishy, it depends on your soil, sandy, loam, clay etc. they all have different saturation and retention capacities.

If you dig a hole, like a post hole, cover it and it fills up every time it rains then you have a good gauge of your water table. Don't rule it out. I'm sure the mines near you are filled with water.

Good luck with this. Be patient, be observant and you will solve this problem.