That looks pretty bad I must say.
The best option for you is to do the following: Solder a new pin header upside down (ie: place it in the holes so that the platic is on top. Solder one of the pins so that you can align the header so that it is straight up. Then solder the pins from BOTH sides. Make sure you hold the iron long enough for the solder to flow through. The problem is that if you damaged the through-plating, it may not conduct from top side to bottom side, so you'll need solder on both side to fix that.
Once you have the pin header soldered properly, remove the plastic, and you'll have perfectly centered pins sticking out of the board.
You'll loose rigidity this way, but as long as you don't keep applying/removing the modchip to the pins, it should hold OK.
For the future, the best way to remove a pin header is to gently slide off the plastic case, apply extra solder to the back (for better heat distribution) and remove it PIN-BY-PIN (use pliers!!!). Then when you have the pins removed, add more solder (for heat distribution again), and suck out the empty holes.
The trick of applying extra solder before removing it seems counter intuitive, but it allows better heat transfer and will help melt all the solder, so that it will come out easier.
Good luck!