We got ours working (shipping via weight method) by:

1.) updating our site's entire Drupal Core
2.) Re-Installing Ubercart and associated modules all at their most recent version.
3.) Enabling all shipping methods - then switching off the ones that weren't needed.

There's definitely something wrong, some conflict somewhere, as this kind of backtracking shouldn't be necessary. We had a stage site, so it could all be done relatively safely, even if it did waste a day or 2.