Where does the 'C' in exhaled CO₂ mostly come from?
CO2 is a product of Cellular Respiration, which generally takes Glucose and molecular Oxygen to produce Carbon Dioxide, water, heat, and allows ADP to be regenerated into ATP (or other various oxidation reactions). The Carbon comes from wherever the acetyl-CoA used in the Citric Acid Cycle came from - either carboyhydrates or fatty-acids (saturated carbon chains).

Simplified reaction:   C6H12O6 (s) + 6 O2 (g) → 6 CO2 (g) + 6 H2O (l) + heat

So, you are correct. CO2 transferring out of the lungs is mostly the result of burning sugars (or fats) for energy (the regeneration of ADP/GDP with respect to human biology).

To that end, the tissues that produce the most CO2 will be the cell-types which constantly require energy. Nominally, muscle tissues. 

Per your comment, broken down fat, or rather, the process of Fatty-Acid Catabolism, results in the production of acetyl-CoA, which is a primary player in the Citric-Acid Cycle. The Citric Acid Cycle, which you should recognize as the Cycle that Pyruvate - the end result of Glycolysis (the breakdown of Glucose into 2x 3-Carbon Pyruvates) - also goes into after being converted into acetyl-CoA by Pyruvate Dehydrogenase.

The sum of all reactions in the citric acid cycle is:
Acetyl-CoA + 3 NAD+ + Q + GDP + Pi + 2 H2O → CoA-SH + 3 NADH + 3 H+ + QH2 + GTP + 2 CO2

So, for a basic breakdown with respect to CO2:


Carbohydrates (Sugars, Starches) → Glucose → Pyruvate + ATP + NADH
Pyruvate → Acetyl-CoA
Lipids (fats) → Lipolysis → Acetyl-CoA
Acetyl-CoA + ... + H20 → ... + CO2