I doubt the manual will give you much to work with, unless you managed to scrounge up a copy of the tech manual, which I think Accurite was pretty tight fisted with, if they even printed one.
Here's my two bits and a non-electrician.
Most of the problems I have ever run into in electronics of that nature that did not work, were mechanical issues. Broken wires, bad solder joints, corrosion, and the like.
If you cannot see any obvious broken wires, etc., look at the numbers on the LED segments and see if they are still available. If you can, find the pin-outs and test to see if the segments themselves are working. Off the top of my head, a 1.5v battery is usually enough to get the individual lines to light. 'Course, you gotta find the ground pin too, and watch polarity. I don't recall ever frying a display segment by reverse polarity, but...
Before anything else, check that the problem does not follow the scale, by swapping the scale connectors and seeing that the scale from the bad segments does work on the good display segments.
But I would likely start by looking for corrosion or a bad connection in the front panel with the segments mounted on it.
If the segments are set up with plugs, and can be swapped internally, that is another tool to cross check whether the problem is before or after the plug.
Anyways, good luck, watch the big ol' capacitors!