Use a TrailGator, never used a tagalong.
Slopping from side to side sounds like the connections / brackets on frame aren't done up tight enough or are out of line. I needed to straighten / nip everything up after the first couple of rides but that sorted it.
I have no problems with sprog slipping off saddle / height of front wheel others have mentioned. Fitted to a Tinbred / old EA30 post and to the lads CNOC16 (lad is a tall 5 year old).
To me, as the Gator works and I've had no problems with it (and have done some relatively rough trail riding) it seems the best value and is much more versatile.
Took the boy up the Planets Trail / Sustrans Route 65 at Easter. Rode from Pluto (?) to the Sun using the Gator then back under his own steam. Couldn't have done this with a proper tagalong.
Each to their own but I'd get a Gator.