You can hire an independent appraiser.
It is a feature on Zillow where you state what price you would be willing to accept on your home from a purchaser.
If you find a home that works for you, you can write an offer contingent on the sale of your current home. This is typically not the type of contingency that a seller would like to see, but with the way the market has been behaving, more and more sellers are willing to look at such offers.
You should contact the local real estate firms in your area and ask for an agent who specializes in your neighborhood. I would interview at least 3 different agents. When you choose an agent, make sure to ask them to put you in touch with a lender so you can get pre-approved right away so there are no hang-ups when you find the place that you want to call home.
You should contact Zillow and have them take it off as a listing. You can also contact a local real estate firm and interview some agents until you find the right fit. The agent can subsequently officially list your property for sale on the MLS.
You could call that agent as he/she might not be aware that this his happening. I have had several calls from clients who have had this same issue.
You should call the local real estate offices in your community and interview several realtors so you can make an informed decision.