Bringing one's own hand sanitizer is pretty common these days. Don't worry, you're not alone.
As the others already suggested, it's best you get the necessary vaccines if you are truly concerned about health issues. Sometimes hysteria can kick in, and let you experience symptoms because of consuming thoughts on health issues. If it give you peace of mind, please by all means do all the necessary health protection that will allow you to enjoy your vacation in India. Additionally, just be careful with what you eat. Whether hygienic or not, sometimes our body's composition is just different. That is, two people (tourist/local) eating the same food may only affect the tourist. The food may not necessarily be bad, but it's just that a foreigner's body may still be in the process of adjusting to the new taste, ingredients, spices, etc. and so may end up sick.