I'm going to assume the links have a split half-way up one side. For more leverage prying a link open, take 2 large wrenches (pipe-wrenches could work) and grab either end of a link. Push in opposite directions, across the split in the link. Reverse the same procedure to close the link again.
It sounds like you could maybe just change which links the carabiner goes through to shorten the chain. Alternatively, what is the bottom attachment like? Can you disconnect the seat and re-connect it higher? You could also consider adding a link/carabiner that jumps several existing links: connect the link below the existing carabiner to a link 5-6 further down.
Swings can involve quite a bit of force, and thus chains need to be fairly strong. If you can't shorten the chain moving the ends, you might have to find someone with proper chain modification tools.
EDIT to try to improve my instructions
(I am not an artist.)
Position each wrench so it bites around the red or green circled area. The wrench handles will be perpendicular to the chain. It doesn't matter if the wrench is slightly loose when force is not being applied (this might be familiar if you've used a pipe wrench before), nor does it matter if you are biting partially on the adjacent link. When you move the wrench handles in opposite directions (again, perpendicular to the chain), the force will work between the wrenches. In many cases you can simply grab the links above and below the one you desire to open; although in practice that might open the adjacent links slightly. NOTE: the links in this image (and in many heavy-duty chains) are welded or fused. These links cannot be opened, they can only be cut.