I've created new dialogue for my follower for each of the follower commands in the "DialogueFollower" quest.

The command "wait here" appears to work, meaning that the option is there and he responds that he is waiting, but he will still follow the player and the option to "follow me" does not show up. I'm not understanding why, because the new dialogue is just a copy of the vanilla dialogue with a new voice file and the conditions set for my character.

Well, once again after days of searching for answers and finally posting a question about it I figured it out on my own.

In case some other poor soul runs into this problem, this is what fixed it for me:

First off, my dialogue was all under a unique quest. I didn't use the vanilla "DialogueFollower" quest. I made a copy of it and copied over individual dialogue choices I wanted to use.
In my naivety I assumed all info would copy over exactly like it was in the original quest so I could simply make my changes and it would be done. What I failed to realize is that the scripts that copied over tried to adjust automatically to the new quest, which almost worked. Unfortunately there are just a couple of things that it didn't get right on it's own.
That would be under "DialogueFollowerScript" under the "Scripts" tab in your quest screen. If you double click on that and bring up the properties for the script you can scroll down to "pFollowerAlias" (and / or "pAnimalAlias" if your follower is an animal).
There you'll see a property with your quest name and below that a blank property (Uh oh!). I was unable to leave my quest name in the first box. I had to change it to "Dialogue Follower" and the second box to "Follower". Obviously for pAnimalAlias you would want it to be "Animal".

This magically solved my issue completely. There may be other problems with some of the other properties, but they haven't surfaced for me yet. Everything seems to be working perfectly.

In the end, I found it better to create a custom quest(s), for my custom follower; rather than amend or copy the big FollowerDialogueQuest.

There is a very good how-to write-up here: http://www.gamesas.com/topic/1349193-wip-an-unexpected-companion/page__view__findpost__p__20583334 (also see the post a couple down from that one ... for a shortcut)

This means I can add more dialogue or action functions to my Follower without hacking around with existing Beth Quests/Scripts or all of the other Follower Variables getting in the way.

Well, if you think about it copying is easier, because you're just writing all the scripting yourself if you do it that way. Here's what I did, and it's a lot shorter than that.

1.) Rename the DialogueFollower and DialogueFavorGeneric quests (click "yes" to make a copy)
2.) Make the changes I mentioned above.
3.) Create each branch that you want your npc to have. Name it whatever you like. I pretty much copied the vanilla, just put my mod name instead of "Dialogue".
4.) Copy a response under that branch. Right click on the copy and choose "Move to topic". Uncheck the "current quest only" box, and choose the quest you made.
5.) Open the response up and change the condition that says "Actor: XXXX" or "Voice Type: XXXX" to your actor or voice type (best to have your own voice type too).
6.) Do that for each response you want your character to have. You can edit the dialogue and emotion to your liking. You can add a voice file. You can add any additional topics you want. It's completely customizable, and you aren't changing any of the vanilla files.

This way you know you're not missing anything. Everything you need is already done for you. You don't even have to open a script (some people aren't comfortable getting into scripting), except for the small property changes above. Both ways I'm sure work well. This just seems easier and more newbie-friendly to me.