If its still sharp you can try lower action and/or less relief.
I always intonate with the old strings. Provided they arent extremely old that is.
The reason for this is a two birds with one stone explanation. Strings dont stretch in, they bend in. Theres no permanent length change whatsoever. All materials have two states of deformation, plastic and elastic; elastic is a deformation from which the material can retain its original shape and plastic is beyond that and permanent. The strings want to be straight, so when you first put them on they take curved shapes over the bridge and nut. They need to be angled in those places; a full plastic deformation to the correct angle, no elastic deformation left. Rather than stretch strings in place, which only does any good by accident, use the handle of pretty much any tool, or anything hard, to force the strings down above the bridge and nut.
This also means that when you put new strings on they are actually in effect orginating from a slightly higher position at the bridge! This affects your intonation (makes it sharp), so press them in over the bridge and see what its like.
I dont know how long it takes for the strings to take their final correct shape, but its certainly a couple of days and you may have see it happen already, and it may be why your intonation is improving.
Everyone else: stop stretching in your strings, start pressing them in!