My grandmother always drank black tea from store-brand or Lipton tea bags, and she always reused them and didn't say anything about the taste.
My current favorite black tea is pu-erh tea from Prince of Peace brand tea bags. I use two tea bags in an extra large tea mug, steep exactly 3 minutes, and use the tea bags three times. I tried using four or more times but sometimes cup #4 is a bit weaker. I honestly don't see much difference between batches.
I have read that the caffeine content goes down in the cups after the first. In fact, in the book "Lose weight with green tea" by Patricia Rouner, it is suggested that steeping a fresh tea bag in hot water for about 50 seconds, then discarding the water removes most caffeine but keeps most of the antioxidants so you can decaf your own tea at home.
I store my used tea bags on a plastic soap dish that has a grid to help the soap dry. This lets the tea bag dry out some between uses.
I might add, however, that if you are drinking tea for the medicinal benefits, you might try using fresh tea bags each time to ensure the maximum antioxidants, and see if that helps your health problem more than drinking tea from reused tea bags.