You may want to take a look at your roof vents (stacks). There's a plastic boot around the vent that cracks over time. Rain water can leak through & travel down the pipe to the slab. If this seems likely, there's something called a Perma-boot (Home Depot, Lowes, etc.) that you can install yourself. It fits over the existing roof vent boot. Good luck!
If the water is coming up from below into the window wells you have a big problem. That issue means your basement is becoming a reverse swimming pool. Imagine taking a big concrete box and putting in in a lake right up to the edge of the box. Eventually many many problems will crop up that your window well issue is just the first warning of.
You are going need to take that water away from the house.
A sump pump will do it quickest and easiest, as long as the power doesn't go out during a storm (gee, somehow that seems like it might happen, plan on having it happen and take what ever measures you think necessary, like back up power for the sump pump).
Redirecting the gutter downspouts will do even better if the power does go out.
Grading the slope of the land around your house to drain away from your house in every direction will also help without power - possibly with buried plastic under the soil sloping away from the house as well as a sort of 'earth sheltered umbrella'.
Adding new better french drains around the building while performing the grading and redirecting the downspouts and adding a sump pump or two would be best of all.
But in the end, if none of the above work, I would recommend abandoning the basement entirely, filling it in with gravel and a sump pump as the water level in your soil will never allow you to have a safely dry basement.
I know one family who had to do this very thing, and was much happier knowing that their foundation wouldn't become a molding swimming pool slowly being crushed by the outside soil/water pressure. It required moving some of the HVAC and other utilities that were down there, and using the space under the floor only as a crawl space.
Best Answer
Your gutter is overflowing because either the gutter itself is clogged up with debris or the down spouts are clogged up.
Your larger sized gutter should be able to handle more rain than the smaller sized items installed on your neighbor's houses. So a smaller size would be even less able to cope with the problem than the size that you have.
I suggest that rather than fitting new gutters it would be more cost effective to unclog the existing gutters and make sure that water can flow properly along the gutters to and then down the spouts.
There is a slight possibility that some of your gutters may have been installed incorrectly such that they have a slope away from the down spouts. If this is the case then rain would gather and run toward the down slope end and then overflow because there is no where for it to go.