"Even though it would be impossible to retain student on a handshake,"
I'm very sorry you don't have any students with a strong interest.
I've never met a dancer who even expected a handshake. Mutual respect gets the job done just fine. Since the interaction between serious teachers and students tends to be paid one lesson at a time except in cases with a lot of personal history prompting bulk payments for convenience, the amount of money that can be lost if someone goes back on their word is small enough that it's worth taking the risk.
These problems really only come about as a result of one or more of three danger factors:
1) Trying to market dancing to a segment of the population that isn't all that interested, and depending on being able to sell it to them in order to eat.
2) Pay in advance packages - this is nothing but a recipe for trouble!
3) Pro/am competition fees so steep that even wealthy clients have to pay over time. Again, a recipe for trouble! If you want to charge that much and you are worth it, charge more for your pro/am lessons to spread out the payment and do the competitions for less.
4) Letting pushy students talk you into business arrangements that are bad for you or bad for them (in the end it's the same thing). If a student makes a counter offer to your usual, say that you will sleep on it and get back to them. Then go discuss it with your mentor, partner, or peers. If the deal is a poor one for either party, in the long run this will fatally poison your relationship - for example as soon as the student's peers or the teacher's peers point out that they got taken for a ride.