Snap Offset won't be helpful for copying/moving partial clip selections as I gather is the case here.
Unfortunately, there is no longer a 'fuzzy' selection option as there was way back in the early days of Cakewalk for DOS when there was no quantizing, and you could only copy entire measures.
I suggest you set snap at a value that's big enough to encompass early notes when drag-selecting, but small enough not to accidentally include notes that belong in the previous measure (maybe 1/16), and then use drag-copy/move with Snap 'By' enabled instead of cut/copy-paste to preserve the tick offset of the selection. I think this would be applicable in Pro Audio.
In SONAR you can also select in the PRV by lassoing (right-click and drag across a time and pitch range) which will select entire notes by just touching them, but I don't think Pro Audio had a Lasso function.