First and foremost, before you do anything, I'd get pre-approved for a mortgage. Since lenders are tightening up on loan approvals, this is the one thing that will either make buying and selling quickly a cinch or a nightmare. If you're pre-approved, you know already you have money to make a deal if something you like should come up. You also know where you stand financially, and how much house you can get. Good news you're debt free, and have your expenses down to the essentials. If you have a good solid income, despite tighter regulations on who gets homeloans right now, you should do well.
Second, after finding out what you qualify for by lenders, I'd secure a real-estate agent, and have them start looking for both potential buyers for your current house and future homes for you to check out, in the neighborhoods you want to move to. While a real estate agent costs money, in the end, if you want a smooth transition, and less of a wait selling your current home, they can help you find buyers quickly, and get you into homes that would otherwise not be listed in more traditonal places like ads, and online.
If your house needs repairs or anything to make it more sellable, I'd do this before the real estate agent sees it. They can always give you ideas on improving your house's sellability too, but you've also got to sell them that they can move your house. So be sure to give them a good first impression. While "staging" is the hot thing with realtors to move a house, be wary if they start talking about putting alot of money into the house before you sell it. You want it cosmetically presentable but at the lowest cost to you as possible. Your goal is to get the most equity out of the sale of your house to get a good downpayment for your next house.
Considering this is more of a buyers market, I'd put your current house up for sale first, and not even start looking for your new house until you sell your current house. No buying until everything is a done deal. Once again, you will be able to move quickly when you do find something becase you'll have your downpayment and preapproved loan.
From your post, I don't think this is your intent, but alot of people make that mistake and wind up getting stuck with several months of two utility bills, two mortgages, cutting two lawns etc. In short, its a financial nightmare. Don't do it. No matter how great of a house you find, unless you make a contingent offer with the seller and they're willing to wait for you to sell your house. Not too likely, but it does happen.
Selling your house first, before finding a new house in this market will not be a problem in the end. Sellers are eager to offload their houses. So, If you need to move fast, I doubt you'll have many people who won't be happy to accomdate a fast turnaround.. just because of the market. Not to mention there is no shortage of good houses available now.. probably at the asking price you're looking for. There really are lots of great deals. Best of all letting one deal pass you by won't be a loss, especially if you don't quite have your financial ducks in a row just yet. I can't stress it enough, Get the cash from the sell of your house and a pre-approved loan first, and the rest will go smoothly. Best of all you'll have more bargaining power so finding that perfect home in this market will be definitely in your favor. Just have patience and think things through carefully.
Last but not least, check with the local housing and development office and find out if there are any special loan or incentive programs for someone like you. There are lots of loan programs for working moms, and maybe even an urban renewal progam in your area, that might provide special help in the sale of your urban home. If you're a minority, many communities offer special loan programs as well. The real estate agent might be able to help with this, or maybe check with city hall's housing office. Good luck.