Can storing my HDDs airtight damage them in anyway?

Solution 1:

First of all "normal air pressure" and "air tight seal" are not mutually exclusive. Secondly "normal air pressure" requirement is just for the operating mode not for the storage. The bottom line is your drives are going to be fine. I mean if they fail - that won't be because you stored them in an air tight container.

Solution 2:

Wrapping using cling film should keep dust out. Keeping moisture out is less likely. I think there will be little effect on moisture in the air INSIDE the wrapping unless it is completely air tight and has been wrapped in a different atmosphere. So, other than regular moisture problems, I suggest that the most important problem is temperature. If the drives are cold, moisture can condense causing a problem.