Given how mind-bendingly stressful home selling and home buying can be, it can be very hard to remember that you're not the first person to ask your question. In fact, it's extremely common.
Or do we find another house, put in an offer contingent upon us selling our house, then sell our house, then move?
Pretty much this. We just went through it a little less than a year ago.
As it was explained to me by my realtor (who is frigging awesome), contingencies are frequently used in real estate.
If the seller is smart, they'll include a "kick-out clause" to go with your contingent offer. Basically, it allows the seller to keep marketing the home while you try to sell yours. If the seller gets another offer, you have a certain amount of time (usually 72 hours) to remove your contingency. Otherwise, the sellers can terminate your contract and go with the new buyers.
If this hasn't been explained to you by your current real estate agent, you need a new real estate agent.
Unrelated to contracts but related to moving: if you're using a moving company,
use a company that will send an agent out to your home and give you an estimate in person! We used a company that gave us an over-the-phone estimate and it was an absolute nightmare.
Long story short: when the bill came due, it totaled 208% more than the
high end of our original estimate. The moving company tried claiming that we misrepresented what we had.
An oversight commission run by the state of Illinois ended up getting involved. The moving company finally took our complaint seriously. They reached out, we settled on an agreement, they mailed me a check, and arbitration was avoided.