Effective Hamiltonians and effective electroweak operators are calculated with the Okubo-Lee-Suzuki formalism for two-nucleon systems. Working within a harmonic oscillator basis, first without and then with a confining harmonic oscillator trap, we demonstrate the effects of renormalization on observables calculated for truncated basis spaces. We illustrate the renormalization effects for the root-mean-square point-proton radius, electric quadrupole moment, magnetic dipole moment, Gamow-Teller transition and neutrinoless double-beta decay operator using nucleon-nucleon interactions from chiral Effective Field Theory. Renormalization effects tend to be larger in the weaker traps and smaller basis spaces suggesting applications to heavier nuclei with transitions dominated by weakly-bound nucleons would be subject to more significant renormalization effects within achievable basis spaces.