Common sense says, to make the trailer more stable, you should lift/level under both axles (or all three) as the lift will be carried by just one axle (spring) otherwise.
When you raise or lower the front after leveling side-to-side it will either increase or decrease the level when you just use one, potentially to the point where it will cause the non-leveled wheel to lift. If the trailer weighs enough, yes it can damage that weight bearing spring.
When we are driving and cross uneven surfaces individual springs are temporarily weighted, but are not normally damaged as the springs are relieved of the weight fairly soon. Parking them that way will eventually flatten the weighted spring.
Not to add more fuel to the fire, but when there is enough drop in the surface you are parking on, it could become necessary to add some leveling between the axles (front to rear.) I have only had to do this twice, but both times it made the trailer more stable, and ultimately, that is one of the several reasons we level the trailers.
I also use a combination of different thicknesses of boards. It makes it easier to fine tune the leveling process. I also use the Lynx levelers occasionally, but only for a major leveling project. If memory serves, the downside of them is that the first block used is 2" and each one added after that adds 1".