What we've found that's effective and not very expensive is to use good-sized plastic boxes and silica gel cat litter (a cup more or less per box) for those dried flowers that need it. You can tie the litter in a cheesecloth bag, if you wish, to make it easier to handle.

Many things don't need any help, even though our summers in Michigan can also be quite humid -- achillea, statice, filipendula, eryngiums, artemisias and many more are stored in our barn in large cardboard bins, open at the top. The most sentitive things are gomphrenas, hollyhocks and helichrysums -- just the stems, in all three cases. They can re-absorb moisture from the atmosphere even after they seem to be quite dry. These are the ones that go into plastic boxes or tubs with lids. We put them in fairly thin layers with each layer separated from the next by a sheet of newspaper. We also put peonies in plastic boxes even though they dry very thoroughly and don't seem to reabsorb moisture, because they are favorite snacks of Indian meal moths. We protect the blooms during drying, then freeze them for a week, then thaw them then store them in boxes with silica gel.

Since we don't have to use a lot of closed boxes and since we sell the flowers, now use aluminum packs of silica gel that can be regenerated again and again by heating the the oven -- we can justify the initial expense that way -- but the silica gel cat litter crystals should be an affordable solution.

I'll also include a link to one supplier of the aluminum packs. In these are often used for storing ammunition, so if you order any you may get on some strange mailing lists.

Here is a link that might be useful: Silica Gel in Aluminum Packs