Odd. I've never had any issues with clogging Vallejo or (Reaper?) dropper bottles. At all. If I've ever had any problems with paint pots/bottles, it has always been with the octangular GW screw cap pots; those were horrible.
I store my paints in a drawer under my painting desk, all bottles standing straight up. So; dark, at room temperature. I still have Citadel paints in the soft octangular bottles with white flip lid that are now close to 25 years old and are still as good as back then.
The only issue I have with Vallejo paints (if that is indeed the brand you were referring to; I have no personal experience with Reaper paints) is that the lighter colours such as yellows (very much), oranges and bright reds tend to clog all their pigments inside the bottle (so not the nozzle), making it one big rattling clump, swimming in what little medium remians fluid. Only the colours I mentioned; blacks, greys, greens etc etc are all still ok, and I have some pretty old (close to 10 years?) Vallejo bottles in there.
What I do though, and I'm sort of anal about this, is after squeezing out the paint I need on my palette, I immediately screw the cap back on, to minimize the time the open bottle (and paint) is exposed to the open air. I don't know if that could be related to your problems, but it works for me. With each paint drop on the palette, from opening to closing, the bottles are usually open for a maximum of 5 seconds , give or take.
As an aside; I do the same thing with my favourite plastic glue; Revell Contacta, with applicator needle. I always heard complaints about the needle drying out, but I (almost) never have that issue, because after applying the glue, I always immediately close up the bottle again, before pressing the to-glue-parts together. This actually helps, as the glue will have a bit of time to start melting the plastic, making the actual fitting/glueing part a bit easier...