Both scenarios are possible and very common...it all depends on what your situation is. For me to give you the best advice I can I'm going to need some more details. I have had clients buy and sell concurrently and then I have had some clients sell first then buy and even vice versa. Send me an email or give me a call so we can figure out what's best for you.
Alex is right he is also in your town and would know condition best. A local expert can help you with market conditions. They will know if homes are selling fast , in that case it could be better to sell first. You may need to sell before you can buy if you are obtaining a loan.
It's certainly possible to do both at once -- as long as you're somewhat flexible. Most areas are experiencing a "seller's market" right now. So the selling part may be easier than buying. But if you're flexible on what/where you're looking, you'll have more options. If you can't find the right house to buy, you may have to find temporary housing until you do.
It's actually quite difficult to do both things simultaneously. It takes time to get a property sold..and even longer to find a suitable one to purchase.
I would not "count on" being able to accomplish this feat, but make arrangements for a short term place to stay after selling and while buying.
I always recommend selling first or simultaniously, but it really depends on your financial situation of course. If you sell your property and can't find a house it will hold up the escrow of you selling! Also is this your primary residence? You might consider renting for a while before jumping into another purchase. If it is an investment, try looking into a 1031 exchange which lets you differ the capital gains tax and still buy another property :-) Please feel free to contact me with any other questions. thanks!
It really depends on your financials. If your income is tight, then there will be a loan contingency that you sell your old home before you can get the loan for your new home. Email me if you have any further questions.