Uh; those "babystays" are actually your lower shrouds. The N-27 does not have a "babystay" which goes from the front lower 1/2 of the mast down about 1/3 forward on the foredeck.
The lower shrouds will help stabilize the mast fore/aft a bit but really they are to help adjust mast rake and/or bend. I had a similar problem on my N-41 and it turned out that the lower diagonal shrouds (lower primaries) were loose. To tighten those I had to go up to the first spreader and tension the diagonals. Don't do this unless you have a good tension gauge like a loos gauge or similar tool.
Your mast is under way more compression while you are sailing so I would not worry about causing the deck to dish with the turnbuckles. But again; be sure and use a good tension gauge and follow the directions for basic rig tune. In addition you should check for changes in rig tension over time. If the shrouds get loose after sailing it could either be stretching or failing shrouds, or compression of the step due to core rot, chainplate rot/failure, or failure of the keel step.
If everything is tight you could still have some harmonic wiggle of the mast depending on wind speed; but it should not shake the boat violently.