I made my own beauty replacement mods, one mod for each race, using various meshes that appear on my computer from mod-downloading. However, I should have done a test go first, because some of the heads or hairs are either incompatible with each other, don't look good, or I simply don't want them. But since I've been using these mods for quite some time, how can I edit or make a new version without causing issues like NPC doubling, or headless people?

I think if I understand your question correctly, the best place to edit your Replacement Pack would be in the Body Parts Tab of the Objects List of TESCS.

If you simply no longer want a head or hair .. just Right click on its name in the list and choose Delete. If you want to say make an Imperial face into a Breton one .. maybe re-ID it if that is appropriate and then choose Breton instead of Imperial in its dropdown list and Save.

If you want to make a whole other Replacement Pack than what you've got now .. open the construction set with your mod selected but nothing made the Active Mod. Then make a new cell and pull the head and hairs that you want from the old pack into the new cell. Then Save and it will ask you to name your new mod. Name this one and Save and now just the heads/hairs you want will be in this new mod for you.