If pressing down gets the thing going, it's quite unlikely you burnt it.
Try to figure out some way to press the part down without putting any force on it sideways, and reheat it.
You'll probably fix it that way.
Anyway, nothing to loose, right ?
Next time you have to solder parts like these, you might consider glueing them to the surface.
Use some glue which can resist heat, doesn't glue instantly so you can adjust / align it and can be removed when needed.
So some silicone based glue would come to mind.
Use a very small amount of glue ofcourse and wait for it to settle before you start soldering.