I am a licensed agent with national certification and a specialty in working with folks, often empty-nestors, who want downsize or simplify or economize in their home costs. Although I am in Rapid City, South Dakota, the answer to your question would be much the same everywhere.
I get this question from my own clients. The best strategy depends so much on so many things unique to the individual's circumstances and to your local market conditions. I'll try to outline some of these factors.
If you have plenty resources to own two homes temporarily, and if home sales are not miserably logjammed right now, then it would probably a lot less stressful for you to buy the new home while still selling the old one. Just keep in mind that with this strategy there is no backspace key. Once you start down that road you need to follow through, even if it means "unloading" your prior residence at a price to make it move (eventually).
Very many people do both. That is they "find" a new home and put an offer on it, contingent upon selling their current home.
From a very practical standpoint, in some markets the sellers will be more flexible in dealing with you if you do not have to sell your current home first. Besides, if you sell your current (more expensive?) home first, then you will have a larger downpayment for the next one. Talk with your lender about temporary bridge loans based on anticipated proceeds of sale from your current home.
With regad to "downsizing." That is a lot less automatic a path to whatever you really are after than many people realize. For example, a smaller home is not necessarily lower maintenance or easier on housework. And it is not necessarily easier on the pocket book each month when paying energy and water costs.
I have found that when a client declares they want to downsize, I ask them to list the 3-5 things that they really are after, as a consequence of downsizing. Is it monthly utilities? Annual maintenance? Less housework? Single-floor mobility for an aging family member? (Maybe you want to assure your adult son or daughter, or an in-law, have no room to move in!) Whatever the reason(s) for downsizing, be sure you consciously list the specific greater goals you expect to accomplish as a consequence of downsizing. Then, when you evaluate homes for sale, it's ironic that you should ignore the size of the home. Rather, make a list of how many of your greater goals would be achieved.
In fact, it is conceivable that your "downsizing" goals might be best achieved by a home that is a tad larger than your current home!
Hope this helps.
Prudential Real Estate
Rapid City, SD