We present high-statistics results for the isovector and flavor diagonal charges of the proton using 11 ensembles of 2+1+1 flavor HISQ fermions. In the isospin symmetric limit, results for the neutron are given by the $u \leftrightarrow d$ interchange. A chiral-continuum fit with leading order corrections was made to extract the connected and disconnected contributions in the continuum limit and at $M_\pi=135$~MeV. All results are given in the $\overline{MS}$ scheme at 2~GeV. The isovector charges, $g_A^{u-d}
= 1.218(25)(30)$, $g_S^{u-d} = 1.022(80)(60) $ and $g_T^{u-d} =
0.989(32)(10)$, are used to obtain low-energy constraints on novel scalar and tensor interactions, $\epsilon_{S}$ and $\epsilon_{T}$, at the TeV scale. The flavor diagonal axial charges are: $g_A^u
\equiv \Delta u \equiv \langle 1 \rangle_{\Delta u^+} =
0.777(25)(30)$, $g_A^d \equiv \Delta d \equiv \langle 1
\rangle_{\Delta d^+} = -0.438(18)(30)$, and $g_A^s \equiv \Delta s
\equiv \langle 1 \rangle_{\Delta s^+} = -0.053(8)$. Their sum gives the total quark contribution to the proton spin, $\sum_{q=u,d,s}
(\frac{1}{2} \Delta q) = 0.143(31)(36)$. This result is in good agreement with the recent COMPASS analysis $0.13 < \frac{1}{2}
\Delta \Sigma < 0.18$. Implications of results for the flavor diagonal tensor charges, $g_T^u = 0.784(28)(10)$, $g_T^d =
-0.204(11)(10)$ and $g_T^s = -0.0027(16)$ for constraining the quark electric dipole moments and their contributions to the neutron electric dipole moment are discussed. These flavor diagonal charges also give the strength of the interaction of dark matter with nucleons via axial and tensor mediators.