Clipless pedals, saddle height and proper starting (and stopping)
The rule of thumb of "the highest possible without wobbling the hips down to reach for pedals" seems like a better rule than "having the barefoot heel just touch the pedal with the leg completely straight when the pedal is completely down".

If your saddle is at the right height, it should be difficult to place a foot on the ground while seated. Now that your saddle is at what seems to be a more appropriate height, you may have to change any habits that developed during the time that you had a lower saddle.

Remaining seated on the saddle at a stop is uncomfortable on a well fitted bike. You should find stops much more comfortable if you stand over your top tube with one foot flat on the ground. With a bit of practice, you should be able to use a single push with your foot to simultaneously begin to pedal forward while lifting yourself onto the saddle. Clipless pedals make this easier because you can easily position the crank by "pedaling backwards" to the desired position. With a little bit of practice, this will become perfectly natural.