I have this issue where some of my clothing will cause others to disappear when I add them to a character.  For instance I'll add a shirt and the shorts will disappear.  Or I'll add a garter belt and stockings and then switch the stockings out for a different pair but for some reason the new pair causes the garter belt to disappear. 
Is it something I'm doing wrong when I save the clothes to my library?
How can I fix this?

Sounds like a layer issue. 
Some of the clothes may be set to the same layer as others. 

For example if I have a shirt that is on layer 5. And try put a jumper on over the top, but that too is layer 5. The jumper will replace the shirt on layer 5 leaving only the jumper. To fix this, you would need to set the shirt to a lower layer number or set the jumper to a higher layer number. 

When you have a piece of clothing on. Select the conform option. Right at the bottom select the cloth layer settings. There you can set the layers. Once the layer is correct you can exit the conform option and save the clothing. It will retain the new layer. 

Reallusion Marketplace Store
Content Store Featured Developer Page

Thanks that was the problem. That process seems like it could be tweaked to avoid unnecessary work.  They should give you a popup that tells you there's a clash and lets you select a different layer.