You are right.

If you download the correct executable installation package (.exe) from Adobe.com (http://www.adobe.com/products/flashplayer/fp_distribution3.html) you don’t need to uninstall the old version, because the installation package is actually an installer/uninstaller.
To check this, you can right-click on the executable installation package and select “properties”: the file description should report “Adobe® Flash® Player Installer/Uninstaller”.

EDITED: the link was incomplete.