Your post brought back horrible memories. DH went through the same thing few years ago. One of the lovely BP medications sent his resting heart rate in the upper 90's. This is a guy who ran everyday for decades. His resting HR was well below 50 when this happened. I don't remember which class of drugs it was anymore but I tossed it out after 1 week. We got him to a specialist and everything went back to normal in exactly 3 weeks. I wished we saved ourselves the headache from the start.
There is a very good chance this is what you are dealing with. You need to contact your doctor and ask her if your elevated HR could be due to the drugs you are taking. You are probably on 1 diuretic and 1 other drug that can fall in any one of many categories. I used to know all the details but not anymore. The diuretic is probably fine. It is the second drug that could be the culprit. You may need to cut it out to see if your HR goes back to normal. Don't do that by yourself. You need to do it under Dr's supervision.
Elevated BP can be due to 2 primary causes, elevated renin a protein secreted by the kidneys (70%) (renin-angiotensin-aldosterone axis) and salt sensitivity (30%). There are some indications that fructose is also a culprit in some people. The nicer one is the sodium salt sensitivity because you can cut out the salt. The renin axis is a lot more complicated because it doesn't just affect your BP. It affects every organ in your body from your brain to your kidneys. There are newer drugs to deal with it but I don't know much about them.
We had DH's renin levels tested with our primary physician. She didn't want to do it but we insisted. I think you need to get that test also to help you decide on the right medicine. DH had the sodium sensitivity variety and that is where we focused our efforts. DH gave up all sodium and all eating out.
I don't know if your doctor went over the DASH diet with you. If not, look it up. Only use goverment sites for info like the National Institute of Health (NIH) and reputable .org. If you want more information, many medical schools have lectures for the public on youtube. I like the ones from UCTV (UCSF medical school). They can be dry but you will find out about the standard of care today.
You need to get to a nephrologist that specializes in hypertension and preventative medicine (as opposed to kidney stones or renal cancer etc...). Look for one at your local medical school before you put yourself through a lot of questionable tests. In the meantime, make sure you get enough protein and at least some carbohydrate to reduce the load on your heart.
Hope you resolve it soon.
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