Johnson's 1812 Dictionary of the English Language gives the following definitions for "Heavy":

1 Weighty; ponderous; tending strongly to the centre.
2 Sorrowful; dejected; depressed.
3 Grevious; oppressive; afflictive.
4 Wanting alacrity; wanting briskness of appearance.
5 Wanting spirit or rapidity of sentiment; unanimated.
6 Wanting activity; indolent; lazy.
7 Drowsy; dull; torpid.
8 Slow; sluggish.
9 Stupid; foolish.
10 Burdensome; troublesome; tedious.
11 Loaded; incumbered; burdened.

The first definition would include the idea of being overweight, but it wouldn't necessarily be so. Just being ponderous would qualify. The other definitions certainly seem to address other areas where Mr. Rushworth may be considered "heavy". I am especially drawn to definitions 4 through 9.

In any event, it appears that JA could have meant a variety of things instead of overweight.