I think i understand the "gap where the locks are" part, but before we get into that - When the door is locked shut does it squish tight up against the compressible weatherstripping on the stops? All three sides?

Yes, it does squish up tight, or at least it appears to (I have to give it an extra push to make it latch). The top and bottom don't seem to be drafty.

It seems to be a problem because there is such a wide space between the door and the entry jamb, so I was wondering what the best way to close that gap.

Thanks for the help.

You may need to reset the entire door, remove the casing on both sides and reshim the door to make it level and square. It may be easier to install a new door and jam.

How is the gap at the top and bottom on the latch side of the door. Does the jamb bow away from the door in the middle? If so, the solution (as another suggested) might be to pull the casing and adjust the jamb. It's really not that hard to do the only problem you'll likely run into this that the casing will hit the wall at a different point requiring some touch-up painting. If you don't feel like doing it yourself any carpenter or competent handyman should be able to do it in less than a hour plus some time to touch up the paint.