So how do I "merge"/connect the clips to prevent this padding from happening? Do I make a copy of the sequence where it's all one big clip?
Padding only happens at the end of exporting. So if you edit the clips one by one in different projects, they all going to have more or less padding at the end. To avoid this, you can put all clips in one project, arrange them the way you want and export only once.
If that doesn't make sense to you I might misunderstood your question. In this case could you upload some screen shot to show how it happens?
Here's a link where you can upload pictures: http://pictr.com/
You can use the URL in your post after uploaded the screen shot.