Inevitably all glass cutters leak oil. I have the perfect solution; don’t put oil in it! This sounds risky as using the cutter without oil will damage it.
So, I still use oil but instead of putting cutter oil inside the cutter, I just saturate a soft cloth in cutter oil and then store it in a tiny plastic to-go container. Before each stroke of the cutter, I run the wheel of the cutter over the fabric once or twice and then cut normally. The oil won’t dry out so just keep it with your cutter until glass accumulates on the fabric. Discard and start over. No more messy storage!