On my landing, similar scenario, vastly lower frequency of vibration. At 1st sign of shimmy, I came full-back on the elevator which (naturally enough) raised the nosewheel off the ground, stopping the shimmy, after which I re-landed the nosewheel with no further problems.
Can't help much on the cure except unloading the nose wheel as you are doing now, but a word of caution - when rolling after landing some light aircraft have enough elevator authority at full aft travel to scrape the tail skid
(even with a forward CG).