There are many different things that can cause a person's hands and feet to feel cold, even when indoors or when it would be unusual for this to happen. As you mention, diabetes is something that can lead to a loss of sensation in the extremities. A condition called peripheral artery disease (PAD) can do the same thing. However, it takes years of uncontrolled diabetes to cause neuropathy symptoms, and PAD is likewise typically a disease of older patients with uncontrolled cholesterol and cardiac risk factors.
In a younger patient, cold extremities are more unusual and so it is best to make an appointment with your primary care doctor
so he or she can go over your entire history and perform a physical exam to try and identify what might be happening. Several possibilities come to mind based on the symptoms you describe. First, thyroid problems can cause people to be either too hot or too cold. Too much or too little thyroid hormone typically also causes several other symptoms, so your doctor will probably ask you about any changes in your weight, skin, and hair, as well as symptoms like fatigue, menstrual changes, or palpitations. Second, problems with the blood vessels in your feet may also cause the symptoms you describe. In some people, the smaller blood vessels in the extremities will sometimes spasm and limit blood flow. This can cause the feet or hands to become very cold or even discolored.
The best way to identify what might be happening with your cold/numb feet is to see your PCP who can then direct you to any appropriate specialists.