Can I soak eggs in something to make them sink into the dye better?
A bit messy, but tested successfully:

Blow out the eggs before dyeing them.

This means you can fill the floating eggs with the liquid dye and they’ll sink. Of course you need to make sure that the dye flows out of the shells again when taking them out.

Speaking from experience, this will work best with eggs that have two holes.

Alternatively you could drill a small hole at the wider end use a syringe with a needle and fill the enlarged air pockets (the reason for the floating observed in old eggs) with a suitable liquid like water.