That wiggling feeling is input from your stoker. As StephenH mentioned, you will start correcting for it and the wiggles will go away. Get enough tandem only miles and your single will feel like the rear wheel is loose
As for upgrades, I would ride it as is for as long as you two can stand then find an older but better quality tandem that may be more worth investing in. Watch some youtube vids on derailleur adjustments and you may surprise yourself with getting them dialed in.
Tandems can be major fun, we just got a second hand Raleigh that I fixed up to ride off road. We are really enjoying our Cannondale road tandem as well.