I've switched to low-carb because LC helps keep away the cravings, but
so
far I have only stopped gaining weight (which is a victory in itself)
but
have not lost any. I may be eating too much. I try to stick to three
meals
per day with no snacks, but I seem to be eating a lot of high-fat
foods. I
know fats are necessary, but is it possible to eat so much fat that
you
won't lose weight even keeping the carbs under, say, 20 grams per day?

Yes. You need to eat less.

Mirek