Have you tried applying the paint directly over the original color? The reason why I'm asking is you might not need any primer.
The main reason to use a primer is to increase adhesion of the finish paint. If the old paint is an acrylic latex (most likely) and you are applying an acrylic latex then technically no primer is needed. The main reason to apply a tinted back ground color is to give better coverage, very beneficial for applying deep or bright colors.
A sage green should cover well on its own, this is an earthy color. You will still need 2 coats but you would be applying 2 coats anyway if using a primer.
Primers don't cover the same way as paint. Their main purpose is to increase adhesion not necessarily cover anything. This is probably what you are experiencing. Try 1 coat of primer then 1 coat of paint and see what that looks like. Now try 2 coats of paint without primer. Do both look good?
I'm concerned about using a pure white primer as now that would need to be covered and this could take 2 coats of paint. This would increase the number of applied coats to 3.
If you feel that a tinted primer is still needed then a neutral milk chocolate background would work well for your gray-green sage color. The only other color I would recommend is a light gray. This is about as neutral as it gets.