If you toss a slip, you lose any items attached to it. So, don't toss them. It's not a leftover message. Storage slips are unique, each with a particular set of data consisting of bits that represent each piece of armor that can be stored on them. When you store a piece of armor with a storage slip, the bit for that armor on the slip itself is switched from 0 to 1, indicating that you have that item in storage. The reverse happens when you remove an item from storage using the slip.

If you bought a new slip and tried to use that to unstore your armor, it would say that you had no armor stored, because all of the bits on that new slip are still set to 0. Likewise, there is no way to store specific data about your armor while it's being stored; only 0, or 1. Thus, anything with an augment or a trial in progress has the trial wiped or is unable to be stored. When you trade your armor with 7/8 seals completed to the NPC, that armor is tossed. Gone. Finito. And your storage slip's bit for that armor is flipped to 1. When you try to get the armor back, it flips the bit to 0 and hands you a brand new piece of the same armor; but, without your trial progress, because the piece with your trial was tossed back when you stored it.

In that sense, no, they are not "safe". If someone is seriously clumsy enough to toss a storage slip, then yes they will lose the armor stored on it. I just keep all of my slips in my Satchel at all times. It's rather impossible to both accidentally take them out and accidentally toss them back to back.