It sounds like a mixture of two problems I think.
Problem 1. Is the consistent 3rd line information attaching to the wrong record?
If so, it should be a group Group Footer rather than an Append template. The symptoms of an incorrect choice will be that the 3rd line gets added to the record below instead of the record above. Every record will be 'out of step' by a line.
Problem 2: Is the intermittent 2nd row attaching to too many records (and probably starting on the wrong record at the moment)?
If so, it should also be a Footer.
Secondly, the normal behavior of a footer (or Append) is that it will attach itself to any records that don't have there own values.
The solution will depend on the version you have. If you have the "Cleared By" option available in your template definition, you can clear the intermittent Line 2 by either the line 3 template or detail template.
If you don't have this option, the advanced field options such as "After Last Field" or "Preceding String" might work. The specifics will be dependent on report layout. Are you able to post a sample?