Relaxation techniques don't work as well as many people claim, in my opinion. There's like Meditation, Deep Breathing, Transcendental Meditation, Progressive Muscle Relaxation, Biofeedback, ASMR, Tai Chi, Yoga... and on and on.
Basically, I think all of these things just trigger the parasympathetic nervous system or relaxation response. It's a short lived response that really can be used for some level of relief, but isn't going to "cure" anxiety.
If you make a plan to do it though, you can trigger the response. Just like lie on the ground and try to breathe slowly without trying to change your thoughts in any way. Eventually you'll feel a change. The change isn't substantial though and works FAR better when you don't have any anxiety to begin with.
As for exercise, it triggers this response (or something along those lines) after you exercise.
When people say to exercise in order to help with depression/anxiety, this is what they mean. Your heart rate is going to be pretty high because of how anxiety affects you, but as long as you don't go way overboard, it shouldn't be a problem. If you do rigorous exercise, you'll get that zoned out feeling afterwards, which is essentially the best you can hope for in terms of relief from exercising. In my experience, that zoned out feeling is preferable to being anxious.
Honestly though, if you feel healthy enough, exercising can be almost as helpful as almost any other form of treatment. You just have to be willing to push yourself consistently and feel healthy enough to do it. Also... it's really cardio you have to do as opposed to weight lifting or something like that.