Is the old version showing up in the 32 bit or 64 bit browser? If it's on the 64 bit one, see if installing the x64 version of 1.6.0_35 fixes the browser plugin.
If this doesn't work, you can launch "Java" from the Control Panel. It will allow you to pick which JRE on your system should be used and lets you pick others manually if they weren't found.
As an aside, for the JVM/JDK used from the command line I like to have more control over what version is being used, so I have a set of system environment variables to set the JVM on the path. I have a JAVA_HOME variable that is included in my PATH set to the bin directory.
I also have a set of system environment variables for each version of JVM installed, e.g.
JAVA_HOME_1_6_35, which I create when I install a new JVM that point to the installation directories of each one.
The JAVA_HOME variable just points to the JDK version I want to use, such as