oof, you might have some trouble with this one, but you can probably do it with a high quality braid (i.e. the small roll that costs ~$25, not the roll that costs ~$10). In this situation you’ll probably want to desolder what’s in between the PCB and the header first, then the pins on the opposite side of the PCB… but honestly it would probably be best to clip the header off, remove the pins individually, then install a new header; it would make the whole process a lot easier.
Sometime braid gets a bad rap but the two most common reasons desoldering with braid can seem like it’s not working is 1.) it’s low quality braid, and 2.) your soldering iron isn’t transferring heat efficiently. Keep you tip tinned and clean throughout, and move your braid along each pin slowly until all the solder is gone; good braid will absorb solder like a sponge. Be gentle with the pressure you use with your iron because you could easily ruin the traces or screen if you blast the PCB with consistent heat. I wouldn’t recommend those spring loaded solder suckers for this, they often don’t work that well and there is a lot of solder to remove in this case.
For your next attempt I would suggest putting the header in place, soldering one pin, then check if the header is level (it probably won’t be perfect). Then you can reheat the pin you just soldered while applying light pressure evenly along the header so that it will shift flush against the PCB. Then remove your iron but hold the header in place for 5 second while the solder cools. Now your header will be in the exact position it should be and you won’t have excess space for solder blobs to drip down like they did here. & you really don’t need a ton of solder, those amount of solder on top of the pins 4-6 in from the right hand side of that picture are perfect, you don’t need any more than that. Best of luck and be patient!