What's likely happening is that you're downloading the program multiple times, saving a .dmg to your desktop each time. Then every time you open the .dmg file, you are creating another Moneydance.app file, which is another copy of the program. Opening the program from different locations could also create multiple shortcuts in your dock at the bottom of the screen. The situation may be even worse than that if you're not replacing your old version of the program.
To install a new version of a program, you should only need to download a .dmg file once. After opening it, drag the .app program file that it contains into your Applications folder, where your programs are stored, replacing the old version of the program.
In your situation, before doing that, I'd suggest removing all .dmg and .app files from your desktop. You should also remove any installations of Moneydance from your Applications folder, as well as from the dock at the bottom of your screen. The reason I'm suggesting this first is that you likely have multiple versions of Moneydance installed at the same time. You're opening different versions without realizing it, which is what is likely causing problems with your data. This will continue happening unless we can remove all other installations of Moneydance.
After you've removed all of these files, you can then download the update from http://infinitekind.com/downloads. This will create a new .dmg file, which you can install in your Applications folder as I described above. When you run the program from there, a new icon should be created in the dock at the bottom of your screen. Hopefully this will help to sort things out so you only have one version of Moneydance installed on your computer.