Serious disease? - The same/similar (though studies have shown this declines within 6 months). Op is correct.

Catching/Transmission? - The current vaccines are systemic and not mucosal, so they don't stop it infecting the upper respiratory tract. - A recent study shows fully-vaccinated households can spread at similar rates to unvaccinated households [1]

[1] https://www.thelancet.com/journals/laninf/article/PIIS1473-3...