Tuners almost never fail to hold pitch. This is a huge guitar player myth. Replace them if they are gritty, or an inconenient gear ratio, or you like the looks of something else better, but not for tuning stability.
Lube up the nut slots, keep the wraps under control, make sure the neck is bolted securly, and only tune up to pitch, never down.
Since you have a fixed bridge, tune up, pull on all your strings, and tune again. That way, any minor binding issue will be avoided, since the guitar is tuned based on a recently stretched string. It will be out of tune the next day, but pulling on all the strings should put it back, or close.
When stringing, go through the bridge, then through the tuner's hole. wrap the string one-half way around the opposite way that the tuner would wind. Pass the tag end under the main length, and pull up and tight. While holding the string taut, wind in until you are near the right pitch, and tune. There should be barely more than a wrap total, but the tension of the string will trap the tag securely.