paint issues tend to be related to adhesion, not tackiness. I've not had issues with vallejo black surface primer on bones personally. washing with soap and water to remove any mold release agents that might still be on the mini. it also gives you time to examine the mini for mold lines that will need cleaning up. i'd be extremely wary of any spirit based paints (like testors enamels) though.
usually the cause of tackiness (aside from color choices (heh)) is propellants and/or solvents in rattlecan primers, paints, and sealers. i would wager the varnishes are likely what's that cause. when trying one, use a sprue of bones material to test on first for any sort of reaction. reaper can talk about reaper paints not having problems - they're a bones paintline that's actually designed to work exceptionally well with bones. there are other threads out there regarding tackiness where people share their experiences with non-reaper paints. i've personally used reaper, citadel, vallejo, and army painter paints without problems. i've used army painter rattlecan primer and sealer and vallejo rattlecan sealer and i've not had issues. i've not had issues using brush on sealer from any of those companies either. come to think of it, i've also tried the citadel rattlecan primers for their contrast line and those were ok for me too.
to play it safe, look at an airbrush. this will let you get the effect of a rattlecan without the problems of worrying about solvents. airbrushes are their own topic, but suffice to say anything you're likely to want to spray through an airbrush is not going to cause issues with bones. badger's stynylrez primers are great. vallejo's mecha line has some airbrush friendly sealers that i've not had issues with, though getting it in the states has been problematic sometimes.
none of this helps you in your current predicament though.
what i recommend for your current state though is soak them in simple green or super clean (purple) for a day or so then have at them with a toothbrush. your goal is to strip all the paints and primers and sealers from the mini. let dry, then a good scrubbing with soap and water. thorough rinse and dry. generally, i'd reseal the mini with army painter sealer to get rid of the tackiness, but it sounds like these guys are pretty well gone. make sure you're letting your paint dry completely and make sure you're letting your sealers dry completely. i usually give the paints a day before sealing and the sealer a day before putting on a second coat if necessary.