I have been keeping BoBs for a couple of years and found these 'plastic' nest box inserts create dampness which leads to mould on the cocoons. This can create 'chalk brood' a disease where the cocoon's larvae hardens into a hard chalk and fails to mature. Better to use the paper inserts.
I winter in a cold room, similar to a root cellar, in metal containers.
The fridge storage will tend to be too dry, using the crisper draw can be better.
I did write up my experiences to date, have a look.

Here is a link that might be useful: Mason Bees

Thanks Beeman!

Are you suggesting I use the paper inserts along with my plastic trays?

If I needed to I could still return the nest I bought for one with wooden trays. (They were just a little more expensive, so I went for the plastic trays) Do the wooden trays have less moisture build up and resulting mold?

Thanks!

I have had very good luck just storing the filled blocks in a screen porch. I am in zone 7A with a last frost date of April 15-30.
I shoot to have the bee blocks relined and hung plus the cocoons in the hatchery box about now. The daffodils are blooming here and the forsythia and cherry trees are also in bloom.
My blocks are the bored hole in wood block type and use rolled parchment paper tubes.
Plastic does not seem to be as good as wood in this case, but I started and have had good success wood so, I have never used a plastic block.
I started with a single 24 hole bee block in 2008 and now have blocks totaling 106 tubes. When filled, I find 8-10 cocoons per tube.
I do have some problem with parasitic wasps burrowing into some tubes, but no mold or mite problems.
Search on my screen name and you will find links to image albums of my blocks. The building details came from a link to an article by Randy Person.