Soldering female header connector to PCB - neighbor pins some how got connected

I'm with Leo. The only thing I've needed flux for is some very old non plated (i.e. bare) wire. And at that I had better luck cleaning than using flux.

My old 0.1" headers solder fine with the suggested solder.

However I've not seen the parts you referenced so I guess I can't comment on what you are /were trying to solder.

I'm quite sure if the OP tries the suggested solder they will have no issues with the parts shown in the photos.