you could do either.
if you want to gel stain, you will have to remove the door from the hinges, and strip off all of the old top coat/stain. sand. once you are at bare wood, stain your new shade.
Some will say you could just do a darker gel coat over what you have. (you still have to sand off the original top coat. that is what has yellowed w/age. oil base clear coats amber w/time) If you just do that, you may end up w/a blotchy type of finish where you have dark and lighter, original stain colors. it will look like crap. I do a lot of wood refinishing. I know what works and what doesn't.
And, if you want to stain them much darker (like a java or espresso) , then you might as well paint them. They're pine doors, not solid oak or mahogany. not a big deal in my book.
painting them will be so much easier than staining them (properly). If you don't believe, practice on one door that you don't see all the time (like upstairs in the corner or something) Or practice on the INSIDE of one of the doors.
here's a color that somewhat matches your gray floor.
change out your old knobs to black while youre at it
a lighter gray color.
or go w/a black door, white trim
you could also do lever handles. This after is similar to your wall color. they did a black
showing how painting the doors in hallway white brightens up everything.
Q