The key to soldering is heat...liquid solder will draw to the heat...as others have said...
don't try to melt the solder with the torch, allow the metal on the opposite side of the torch to get hot enough to met the solder and it only takes a small amount of solder.
When replacing sections...sometimes it is easiest to take a small file/blade and remove the "stops" inside of a copper coupling...this allows you to slide the couplings above and below the cuts...then cut your section in, flux everything and slide the couplings so they center on the joints...then solder
Couplings are sold with no "stops". Ask for a repair coupling.