My gyno said to use ibuprofen, or a painkiller that is also an anti-inflammatory, and to take it as soon as you feel the slightest twinge of a cramp, and take it every hour or two hours as directed for however many days you usually have cramps. Then she told me if the cramps do not subside by the end of the first day of cramping (and I absolutely DO NOT recommend this unless you have yuor doctor's permission and supervision) to take up to 800 mg at a time. I don't do that, though. I'm on BC now and rarely get cramps.
Do not take aspirin, by the way. It thins your blood.
And yes, tampons would probably be a cause for worse cramps. I know they do that to me. It makes sense, too, because most of the time cramps are caused by the swelling of various the uterus, which causes it to press against pretty much everything else. So sticking a wad of cotton up there would probably create more pressure. I also found that tampons made my entire pelvic area hurt, as well as my upper thighs.
If your cramps are really bad, you should really see a gynocologist. They offered me prescription-strength painkillers (which I declined because the BC has been helping me immensely) and free samples of those single-use heating pads.
"Anyone who knows anything of history knows that great social changes are impossible without feminine upheaval. Social progress can be measured exactly by the social position of the fair sex, the ugly ones included." Karl Marx