That trace above the fifth hole from left on the second photo looks damaged with possibly two more on the top pic (leftmost and rightmost 'dangling' trace on the upper row). Wouldn't be too optimistic about this one without some repair...
In any event you need to suck the holes with solder still in them (not the NC ones of course) clean before you install the new module. Also, use a socket, particularly as I wouldn't be confident this board has survived the abuse, and even if it has you never want to expose a board to this more than once.
I'm no old hand when it comes to soldering, but I succesfully managed my first RTC by this technique:
- slather the area in flux
- put a little drop of new solder on every pin
- heat well to melt new into old,
- when it's all hot & runny, suck it out with a solder sucker
- rinse & repeat for all the holes
If (as inevitably happens) some holes aren't clean, new blob of new solder, heat it all and suck again.
No violence! Otherwise stuff breaks...