Yes, individuals with diabetes may donate blood as long as their diabetes is controlled. This is true regardless of the type of medications used to control blood glucose, except anyone who may have used beef insulin (specifically that which was manufactured in the UK) since 1980 due to the risk of "mad cow" disease. You will liekyl find this link helpful: http://www.redcrossblood.org/donating-blood/eligibility-requirements/eligibility-criteria-topic#med_cond
Elevated blood glucose can cause stomach upset and nausea in some individuals. You are specifically asking about "continued" nausea without giving us any information about what caused it in the first place. I recommend that you talk to your health care provider about your concerns.