If your action is pretty high, that could cause some problems measuring intonation by fretting at the 12th. Have you tried intonating to the harmonic at the 12th? I don't like that since it's not reflective of your actual playing, but will show you that the strings CAN be intonated. The only reason they wouldn't be able to be intonated,
AFAIK, is if the bridge was not positioned correctly.
Also try fretting the note at the 12th with the same pressure you'd use when playing.
Also, the generally marketed tuners aren't as precise as, say, a strobe, but you should be able to get closer results than you seem to be seeing.
And, yes, it's normal for the pitch to change as the note tails off.
Try a few more things, then post your notes. Consider resetting that bridge saddle all the way in the other direction -- so you can confirm that your adjustments are making measurable changes, and eliminate your process from the possible causes of confusion.
ltt