You don't have a keel, but you can use the suns gravity with a similar goal. The sun's gravity is much stronger than a sail's thrust so you can't hover, but you do end up in a higher orbit by aiming it at the sun. However, this averages over an orbit so you don't gain energy over time.

But, aim the sail off-center say 45 degrees from the sun and you can pick if that's adding or subtracting angular momentum from your orbit over time. Reduce angular momentum and gravity pulls you into a lower orbit and thus more orbits/year. Reverse that and your gain angular momentum and get fewer orbits per year. L1 is a special case as it's balanced between the sun and earth so you have the same number of orbits per year.

Now, rather than just orbiting the sun your also orbiting the earth. But, similar manovers work just fine.

PS: It's even more complected as you end up in elliptical orbits if you don't keep things balanced etc, but that's the basic idea.