No, J&J and AZ are not dead/weakened COVID viruses, they're inactive adenoviruses carrying the DNA, which our cells use to make the CV spike protein antigen to stimulate the body's immune response to that antigen, so that if the "real" CV infects the host, the body is prepared with antibodies and T/B cells to fight the virus. The mRNA vaccines short circuit this loop by delivering the actual mRNA to trigger production of the virus's spike protein.
For all three vaccines, they're simply delivering genetic material our cells use to make the spike protein, triggering the immune response, rather than the weakened/dead virus approach, which essentially delivers the actual antigen (the spike protein on the actual virus) to trigger the immune response. The DNA or mRNA simply drive the immune system response - they are not involved in any kind of mutagenic response (that's been tested) in the body at all, so cancers would not be expected.
Furthermore, these new vaccines are on the verge of becoming the most closely studied vaccines in history, with tens of millions and now hundreds of millions of people having been vaccinated in just months - and this rare side effect in about 1 in 1MM people is actually strong evidence that the vaccine safety monitoring system (VAERS) is working nearly perfectly. Clinical trials with 20-40K people can only "see" side effects that are present in 1-2 per 40K or so, which is the equivalent of 50 per 1MM, so we should consider it a huge positive that we can now see side effects in the range of 1 in 1MM. IMO, this should actually give people even more confidence in getting these vaccines.
Lastly, explain for me your calculus here. I think you're close in age to me (50-64), meaning you're chances of dying from COVID are roughly about 4300 per 1MM (~90K dead in ~60MM people in that age group, of which about 35% have likely been infected or 21MM) if you become infected (much higher death rate if you get symptomatic COVID) and your chances of being hospitalized are about 3-4X that or about ~15,000 per 1MM. On the flip side, your chances of being hospitalized if you from COVID if you get vaccinated are very close to zero and your chance of dying are zero, while the risks of being vaccinated are at most a serious side effect in 1 in 1MM people (if this clotting side effect is tied to the vaccine). I simply don't understand, looking at the comparative risk/benefit of the vaccine, how anyone in our age group (and anyone, really, as these comparisons are still way positive for the vaccine even for 20 year olds) would not jump at getting accinated.