I would say that "movements in my head" and dizziness are not common responses. Check your breathing. Holding your breath, especially during exertion can spike your blood pressure. Making sure you continue to breathe may help.
@Dave is correct that you should consult with your physician about your exercise program, especially if you have a medical condition such as high blood pressure. Your doctor will give you safe guidelines. When you go to your doctor, be prepared to explain your symptoms in response to your weightlifting and pushups. Ask specific questions regarding your workouts and what blood pressure readings are safe for you.
This Mayo Clinic article gives some parameters for blood pressure readings and weightlifting. For example, they suggest:
If your blood pressure is between 140 to 170 mm Hg systolic or 90 to 109 mm Hg diastolic, check with your doctor before starting a weightlifting program to discuss any precautions or special considerations.
This article gives HBP symptoms that require immediate medical attention. However, HBP is often without any noticable symptoms. It is important that you understand your condition and how to exercise safely for better health. Also ask your doctor about your diet. Good luck.