You can buy a small tube of powdered graphite at a hardware store for about a dollar. Apply some to each string slot on the nut when you put each new string on. I also put some on the saddles where the string touches but I don't know how much this helps, but I do it. This helps tremendously to prevent string binding at the nut.
I also do not prefer DR strings because of tuning problems.
I put a direct drop in set of Grover 406c mini locking tuners on my PRS SE Singlecut Tremolo guitar and they work great. I think they are the best locking tuners I have and they are automatic, no knob to turn or come loose.
Since I installed the drop in locking tuners on my Singlecut Tremolo it stays in tune incredibly well. It "seriously" stays in tune for long periods of time and drifts only slightly when it does.
You have to break in new strings to get the slack stretched out of them. I'd try some different strings from a major company and stay with nines at least for now until you get tuned into the guitar. Make sure the strings are seated good in the bridge at the barrel ends.
You should have this worked out without any major problem, even without locking tuners. The stock tuners on my new SE Santana work great but not as steady as the locking ones.