If you feel confident in adjusting your own truss rod then that should be checked first. Hold the low e string down at the first fret and the 15th fret, in the middle between those point the string should be just barely above the fretboard. If there is a lot of gap, more than a 64th of an inch, then the truss rod needs a little tightening. If there is no gap then the truss rod is too tight. Always loosen the strings to adjust the rod, and turn it only a quarter turn at a time.
After adjusting the truss rod let the guitar sit for a few minutes then tune it to pitch again and recheck the height of the string between the 1st and 15th fret points. Repeat this process until you get a slight gap in the middle.
As far a the Floyd, once you get the truss rod right, with the guitar tuned to pitch measure how high the strings are at the 12th and the 20th frets and post the measurements here. Then someone can probably tell you if it's not right. Good luck with it. Post back if you have any more questions. :icon_thumright: