There are alternative options to a direct 1031 exchange where you will still have debt on the books, but you can reduce the risk of mortgage payments and increase your cash flow.
The primary transaction is called a 1031- TIC Exchange where you sell your property and exchange the equity and debt into a fractional ownership of a larger property. You do this through a 1031 TIC syndicator. The TIC syndicator buys high cash flow properties with long term leases. Essentially what they do is they collect rent from the tenants, use that to pay down the debt, and send the excess rent to you the investor. Cash flow yields vary, but 6.5-7.0% is about right.
Be careful about what syndicators you work with as they vary widely. Focus on the property first (a syndicator is only as good as his product). Look for buildings in stable, growing markets, with long term leases from high credit quality tenants.
I've done a fair amount of research on this topic and posted my sources below, but quick index of the sources:
-the 1031 tic exchange described above and the slightly more esoteric 1031 to a REIT: http://www.nestegginvesting.com
-how to evaluate a tic sponsor: http://www.1031-tic-exchange.net
-Performing a 1031-121 on your primary residence: http://www.1031-tic-exchange.info/
-1031-tic listings: http://www.1031research.com/