I would recommend that you go to see your primary care doctor
about this, especially since the symptoms have been going on for a while without getting significantly better.
It is certainly true that heavy caffeine consumption can cause a muscle tremor, but this would not usually be limited to the thumb and you would typically notice some improvement already since you have cut down on caffeine.
Base on your description of how you spend your time working at a computer, I have a strong suspicion that this might be due to fatigue of muscles in the thumb or perhaps a repetitive motion injury/strain from using the computer all day long. Your doctor will be able to examine for evidence of swelling, pain, or weakness. They'll also be able to make recommendations about how to make this better. This might include using an ergonomic keyboard or limiting your time at the computer. It might also involve ice application to the area, wearing a brace, or taking anti inflammatory medications, such as ibuprofen.
Start by making an appointment with your doctor at your earliest convenience, and I hope that you are feeling better soon!