I can't say without knowing the specific ingredients and solvents, but just in general.

Different ingredients used for coloring dissolve differently in various solvents. Unfortunately, colors that are not from single ingredients sometimes have to be made from ingredients that don't dissolve well in the same solvents. Also, different solvents and ingredients are less shelf stable than others or react more with oxygen or light.

Generally, if you want to keep pigments, or anything really, from decomposing, you need to store them in the dark or a totally opaque container and in a vacuum or in a very well sealed container, preferably topped off with an inert gas, or at least not oxygen which is very volatile.

The alcohol or chemical smells usually are the solvent evaporating. Things that harden in without special additives or things like UV do so by the solvents evaporating. The quicker the solvent evaporates, the quicker the paint sets. Alcohol evaporates much faster than water, for example.

Anyway, the containers that cheaper paints come in, generally are not 100% air tight or light blocking. (Remember, many thin plastics are air permeable, just barely, air particles are way smaller than water particles so something can be water right but not air tight.) Often I put my paints and similar things I dont use often in a vacuum bag that you cam pump the air out of and store them in a thick, dark box.

Acrylic paint uses water as the solvent, not alcohol, so that's probably not what you're smelling. My guess would be the binder breaking down, which is why it turns into a thin fluid while retaining its color. Hard to say for sure without more details like a list of ingredients.