You have 2 beautiful screw studs to mount a thermal gaurd plate, I would fit a sheet of metal mounted on them, and kapton tape between the board and guard to prevent any cross flow of heat,
Preheat the board to something like 130C, even better if you have something from underneath to keep it toasty,
Then just go overkill on the existing connectors, melt them off if you have to,
Once removed, clean back the board with wick, tin any pads, lay down some flux, get the area hot enough that the solder begins to flow on the pads, switch to heating with the hot air from the underside, and place down a new replacement socket, nudge it lightly until you see it drop on both sides, give it a few more laps with hot air underside with some more nudges, then you should be set.
For those bigger center connectors, as long as you keep to one of the edge connectors at a time they should be fairly steady as the other half of the connector should be holding it in place.
Edit, OK now I see the 2 giant connectors on the other side, this makes things harder. when it comes to direct underside heating... your going to have to go a bit warmer than I would normally on the board, and may be worthwhile preheating the new connector before placing it, this way it doesn't immediately freeze the joints.