You don't need a dual band adapter as long as you have a good connection and not too much noise on the 2.4Ghz band.
Most dual band routers actually broadcast 2 separate wireless networks, one on the 2.4Ghz band and another on the 5Ghz band. 5Ghz often has less range and is impacted more by walls, etc. But there's much more limited space on the 2.4Ghz band and in crowded areas you can sometimes find your throughput plummets because your devices are competing with all those around them.
The "single band" adapter will only connect to 2.4Ghz wireless. Any 5Ghz wireless is not an option for you. Having said that, if you're on 2.4Ghz now and you get a nice reliable connection, there's nothing to worry about.
Ethernet cable is always best if you possibly can, but a nice, strong wireless signal is just fine as long as you don't have too much competition.