Contrary to popular belief, the most difficult negotiating point is often not price. It's date! I agree with the answers below that encouraged you to acquire your mortgage pre-approval first. Ask a lot of questions of the lender. If the lender says that you can afford to buy before you sell, YOU need to decide if you WANT to afford it. There are other options available as well (i.e. bridge loans, etc). Yes, you will need the 10%, but also closing costs, unless the lender will finance them as well.
In making the decision whether to sell or buy simultaneously or not, you'll need to consider the average marketing time PLUS approval time for your building/neighborhood. The broker/agent you choose can and should help you with that based on your market analysis and building stats.
Simultaneous closings happen. They require a lot of orchestration between all the parties (sellers, buyers, attorneys, agents). Make sure you choose an agent who specializes in buyer representation as well as seller representation. I have worked with many clients over the years who have needed simultaneous transactions for financial reasons.
One option for those who can't orchestrate same day transactions or are afraid of or can't afford two mortgages (I understand you don't have a mortgage) is temporary housing. Sell first. Move in with family or friends. Then buy.
Without sounding like a sales pitch, I'd be happy to discuss your options with you. While I am very aware of the current market and do my best to educate my clients, now is a great time for savvy buyers and sellers.
It would be my pleasure to speak with you and review your options. I can be reached at 646-442-7180.
Good luck to you.