This is not uncommon on older saxes. As Paul? said swedging is often a solution that works. Unfortunately that is something that requires the special collet tool or swedging pliers. In some cases there is not enough "tube" showing on one side of the key to do an adequate swedging job. One side is tight, but the other is loose. In cases such as this, a solution is to make and fit an oversized rod for that specific key. Which is not as hard as it sounds, but again it takes special tools to do it.
The right to left wobble in palm keys needs to be corrected, because every time the key comes down in a different spot it is going to leak. When there is a leak like that in the top of the horn, then nothing works well below it.
Another alternative is to replace the short section of hinge tubing on this key. There are multiple fixes that will cure the problem that don't involve replacing the key. Don't forget, the replacement key you may find will quite likely need fitting as well and may be in similar or worse condition than your original.
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