Aspartame is metabolized into methanol which breaks down intk formaldehyde which is carcinogenic. The quantities required to actually produce a significant amount compared to what our bodies produce naturally would be very high, so like you said, you would need to drink significant amounts 10s-100s of cans per day) to reach a level consistent with significantly increasing your risk of cancer.

There may be other side effects on health depending on what type of artificial sweetener you use such as triggering an insulin response without actual blood sugar increase, resulting in increased insulin tolerance, so limiting yourself is still a good idea but it is likely a better alternative to normal sugar.