Were there any Jedi who left the Jedi Order, and continued using his or her lightsaber and Force abilities for whatever they deemed necessary? Specifically, were there Jedi who went renegade but did not turn to the Sith?
Almost all known Dark Jedi in the period between the death of the Emperor and the Yuuzhan Vong invasion were not actually Sith. Joruus C'baoth is the quintessential example, but there are others, such as Kueller and Brakiss.
Darths Tyrannus, Caedus and Krayt did not turn to the Sith until years after they abandoned the Jedi Order.
Mara Jade left the Jedi Order without falling to the Dark Side, though she later returned to the Order when she married Luke Skywalker. Raynar Thul was left isolated from the Order when his ship crashed in the Unknown Regions, yet did not fall to the Dark Side or become a Sith.
There are other examples, some of which have been given above. It was entirely possible for a Jedi to abandon the order and no become a Sith, even if they fell to the Dark Side.
One does not have to turn Sith just for not wanting to comply with the Jedi rules.
There are many things to consider. First of all, there are the so called Grey Jedi that have the purpose to maintain balance in the Force. (category 1) Then there are exiles that want nothing to do with the Sith or Jedi. (category 2) As we will see in the next expansion of SWTOR (trailer here:), some Force-trained are totally separate from Jedi and Sith and hunt both of them (category 3).
Also, alignments are more complex than just being Sith or Jedi.
But on top of all else, both Sith and Jedi ways are limited by their own doctrines to only half of the possibilities that exist in the Force. In order to have true understanding of the Force, you need to know both sides and transcend their synthetic limitations (one given by lawful rules that bind and one given by chaotic will to have unlimited power).
Ahsoka Tano was expelled from the Jedi Order and put on trial for a crime she didn't commit. She was eventually acquitted and pardoned by the Jedi Council, and she was offered to be reinstated as a Jedi. However, she decided to leave the Jedi Order. Since she was no longer a Jedi she survived Order 66 and later became a rebel against the Empire.