The answer to this unfortunately involves talking about cardiac physiology.
Blood pressure is a combination of cardiac output and your systemic vascular resistance of your arterial blood
vessels. Cardiac output is made up of your heart rate
multiplied by your stroke
volume (the amount of blood that is ejected per beat). Your stroke volume is effected by something called venous return. Venous return is the blood that flows back into your heart from the venous blood
Definitions over, on with the explanation. Your sinus tachycardia
will cause your cardiac output to fall. This is because your heart is beating to quickly to let it fill in between each contraction. Your blood pressure will fall as your cardiac output is low. You faint because you have reduced blood flow to your brain.
When you hit the ground your venous return will increase as more blood will be available to your heart. When you are standing the blood in your legs finds it hard to get back to your heart, but when you are lying down it doesn't have to overcome the force of gravity.
Your veins pulse because you have an increased venous return.
In summary, you will have hypotension
when you faint and your pulsing veins are not an indication that your blood pressure is high but that your venous return is.
Incidently the drugs that you are taking for your hypertension
may not be helping. It might be worth seeing your GP to find out if these are still necessary.