There’s two types of diabetes, Type 1 and Type 2.
We seem to be confusing them a lot here.
Type 1: Usually genetic, perhaps caused by some deficiency in the immune system, which destroys pancreatic insulin-producing cells. Type 1 diabetics do not produce their own insulin (at least not in sufficient quantities). As such, they cannot process glucose into glycogen, and will eventually suffer from hyperglycemia and die without administering insulin to regulate their blood sugar levels.
This cannot be managed through diet or exercise, to my knowledge.
Type 2: Usually happens as a result of diet and/or old age. Type 2 diabetics produce insulin normally, but are resistant to it, and thus may need to take extra insulin to control their blood sugar levels in some cases. They should also regulate their diet and exercise to avoid over-reliance on insulin injections, and the condition can be controlled entirely without administering extra insulin, in many cases.
In-game, this means that a Type 1 diabetic would be screwed. No two ways about it. Without refrigeration, there would simply not be sufficient quantities of usable insulin around after the apocalypse to keep them going.
As for a Type 2 diabetic, I guess it depends on how carefully we want to simulate the condition. I think a trait requiring the player to avoid excessive blood-sugar levels (i.e. avoid chocolate and sweets, and perhaps avoid getting sated too often) would be reasonable, perhaps with a termporary hyperglycemic condition associated with failing to control diet.
If we wanted to ignore the refrigeration of insulin, then I guess we could have both types, but they are very very different, and a Type 1 diabetic would require far more insulin than a Type 2 diabetic, not to mention a blood-sugar monitor and a supply of clean needle-heads.
