58 year old female, bursitis in shoulders and knees flares up when I rigorously run or cycle for more than twenty minutes to lower cholesterol. What?

Choices limited. You can try resistance training (ie weights). I would suggest low weights with high reps. Other possibilities include aerobics classes or even water aerobics. Looking at your situation from another perspective, maybe it is time to have your conditions evaluated for treatment if their symptoms limit your recreational and fitness activities.