I'm no expert on STP since I use it once a decade, but from memory...
STP is rarely needed. If you use it, there's a special male connector. That goes into a device with a special female connector that grounds it. You ground both ends IF they have a common ground, only one end if you don't. Ask me how much money I've made replacing shit when people grounded a cable on two ends with separate grounds, and lightning induced voltage into one end (even though the building wasn't hit, it was just in the air). An attic should never have STP, that's crazy. Oh, and in outdoor applications you ALSO need a surge arrestor with its own ground. If the device is PoE, then obviously you can ground both ends.
My last run of it was almost ten years ago, to go around 450 feet (yes, out of spec, it worked) under a boat dock. The far end was specifically ground tested to be sure it was equivalent to the head end, and we grounded both.
Again, no expert on STP, but I think if you have existing ungrounded runs of it, you're going to be fine. I don't think there's a need to remove it.