Recent nuclear magnetic resonance studies [A. Pustogow {it et al.}, arXiv:1904.00047] have challenged the prevalent chiral triplet pairing scenario proposed for Sr$_2$RuO$_4$. To provide guidance from microscopic theory as to which other pair states might be compatible with the new data, we perform a detailed theoretical study of spin-fluctuation mediated pairing for this compound. We map out the phase diagram as a function of spin-orbit coupling, interaction parameters, and band-structure properties over physically reasonable ranges, comparing when possible with photoemission and inelastic neutron scattering data information. We find that even-parity pseudospin singlet solutions dominate large regions of the phase diagram, but in certain regimes spin-orbit coupling favors a near-nodal odd-parity triplet superconducting state, which is either helical or chiral depending on the proximity of the $gamma$ band to the van Hove points. A surprising near-degeneracy of the nodal $s^prime$- and $d_{x^2-y^2}$-wave solutions leads to the possibility of a near-nodal time-reversal symmetry broken $s^prime+id_{x^2-y^2}$ pair state. Predictions for the temperature dependence of the Knight shift for fields in and out of plane are presented for all states.
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