Regarding the deuteron ($np$), the concern is isospin. In the strong nuclear force, neutrons and protons are the isospin down and up states of a single particle: the nucleon. Thus: $n$ and $p$ are identical particles, and need to be in an antisymmetric state:
$$ \Psi(n_1, n_2) = \psi(\vec r)\chi_S\tau_I^{I_3} $$
where
$$\psi(\vec r) = f(r)Y_0^0(\theta, \phi) $$
is symmetric S-wave. If the spin wave function is an antisymmetric $S=0$ singlet:
$$\chi_0 = \frac 1{\sqrt 2}(|\uparrow\downarrow\rangle- |\downarrow\uparrow\rangle)$$
then the Pauli exclusion principle says the isospin wave function is a symmetric $I=1$ triplet:
$$ \tau_1^1 = |pp\rangle$$
$$ \tau_1^0 = \frac 1{\sqrt 2}(|pn\rangle+ |np\rangle)$$
$$ \tau_1^{-1} = |nn\rangle$$
Since $ \tau_1^1$ ($^2$He) and $\tau_1^{-1}$ (the di-neutron) are not observed, we assume the deuteron ($^2$H) is iso-singlet ($I=0$):
$$ \tau_0^0 = \frac 1{\sqrt 2}(|pn\rangle- |np\rangle)$$
and hence the spin state is $S=1$, and indeed, the deuteron is spin 1.
The two-nucleon potential is complicated. Perhaps the most famous version is the Argonne V-18 potential (https://www.phy.anl.gov/theory/research/av18/), named because it was developed at Argonne National Lab, it is a formula for $V(n_1, n_2)$, and it has 18 terms.
The various terms can be attributed to effective field theory processes such as one-pion exchange (the pion acts like a pseudoscalar exchange boson). This one term looks like:
$$ V_{\pi}=V_0(\vec \tau_1\cdot \vec \tau_2)\big[
\vec \sigma_1\cdot \vec \sigma_2+S_{12}\big(
1+\vec 3m_{\pi}r+\vec 3(m_{\pi}r)^2
\big)
\big]
\frac{e^{-m_{\pi}r}}{m_{\pi}r}
$$
where $\vec \sigma_1\cdot \vec \sigma_2$ is the dot product of spin operators (a scalar operator) and
$$S_{12}=2\big[3\frac{(\vec S\cdot \vec r)^2}{r^2}-\vec S^2\big]$$
is a (non central) tensor spin operator. ($\vec S=\vec \sigma_1 + \vec \sigma_2$ is the total spin).
The isoscalar operator is the dot product (in isospin space) of the iso-spin operators:
$$\vec \tau_1\cdot \vec \tau_2$$
(It's called isospin because the math describing it is identical to spin 1/2 particles.) Note also, that the pion is an isovector triplet:
$$ \pi^+ = |u\bar u\rangle$$
$$ \pi^0 = (|u\bar d\rangle - |d\bar u\rangle)/\sqrt 2$$
$$ \pi^- = |d\bar d\rangle$$
(The minus sign is because of antiquarks, not because the state is antisymmetric).
...and that's just the pion. There is also $\omega$, $\eta$, $\eta'$, $f_0$, $\rho$, $\sigma$, kaon exchange (that includes strangeness), and more. Other terms include spin-orbit ($\vec L\cdot \vec S)$, sigma minus p ($(\vec \sigma_1\cdot \vec p_1)(\vec \sigma_2\cdot \vec p_2)$), exchange forces $[(\vec \sigma_1\cdot \vec L)(\vec \sigma_2\cdot \vec L) + (\vec \sigma_2\cdot \vec L)(\vec \sigma_1\cdot \vec L)]$, for example.
The takeaway here is that the spin and isospin nature of the two nucleon force cannot be simply described as "the spins like to be aligned". Another takeaway that may not be obvious to non-experts (esp. b/c of the pictures we see of nuclei) is that protons and neutrons don't retain their identity. In a deuteron in the $S_z=0$ state, not only is one of the particle in a mix of spin up and down, it's also in a mixture of proton and neutron.
When you start looking at nuclei, you now have to consider three and four nucleon forces, which are novel for many physicists. The three nucleon force, for example, is not the standard 3 body problem with two-nucleon forces between all particles. It's that, plus a new forces that can only be considered as a 3-body interaction (https://en.wikipedia.org/wiki/Three-body_force).
There is also the EMC effect (https://en.wikipedia.org/wiki/EMC_effect), which is observation that quark structure functions of the proton and neutron are modified in the nuclear environment...that is, protons in nuclear environments may be different from free protons.
And that's all effective field theory. A fundamental QCD based description is a long way off.
