I'm not certain what exactly you're looking for with your first bullet, but I'll address the second bullet by looking a an exactly-solvable model with a nontrivial RG flow which can be determined just by looking directly at correlation functions rather than deriving beta functions. It's the large-$N$ limit of $\phi^4$ theory:
$$
\mathcal{S} = \int d^dx \,\left[ \frac{1}{2} \left( \partial_{\mu} \phi_{\alpha} \right)^2 + \frac{\lambda}{2N} \left( \phi^2_{\alpha} - N m^2 \right)^2 \right].
$$
Here I'm using slightly different notation to facilitate a large-$N$ expansion, but by multiplying out the last term you get the usual $\phi^4$ theory with some factors of $N$ and $m^2$ placed differently than you're used to (and an unimportant constant). For $2 < d < 4$, the RG flow of this theory is known to look like the following (picture credit https://arxiv.org/abs/1811.03182):
The flows to $m^2 = \pm \infty$ describe a flow to a gapped theory where all correlation functions decay exponentially at long distances, so they're not quite as interesting as the line connecting the free massless theory $G$ to the massless Wilson-Fisher fixed point $WF$. We will see that although all correlation functions will be algebraic at both large- and small- distances, there is a crossover of critical exponents between the two asymptotic cases. The constant $\lambda$, which has units of $\mathrm{(energy)}^{4-d}$, will play an essential role here.
I won't go into every detail of the large-$N$ solution, but I'll outline it (I'm essentially following Section II of this paper, which itself using a similar method as Polyakov's textbook). The first step is to use a Hubbard-Stratanovich transformation to write
$$
\mathcal{Z} = \int \mathcal{D}\phi \, e^{-\mathcal{S}} = \int \mathcal{D}\phi \mathcal{D}\tilde{\sigma} \, e^{-\mathcal{S}'}
$$
where
$$
\mathcal{S}' = \int d^dx \,\left[ \frac{1}{2} \left( \partial_{\mu} \phi_{\alpha} \right)^2 + \frac{i \tilde{\sigma}}{2\sqrt{N}} \left( \phi^2_{\alpha} - N m^2 \right) - \frac{\tilde{\sigma}^2}{8 \lambda} \right].
$$
At this point, we can integrate out the Gaussian fields $\phi_{\alpha}$, obtaining a theory of the form $\mathcal{Z} = \int \mathcal{D} \tilde{\sigma} \, e^{- N \mathcal{S}[\tilde{\sigma}]}$. This can be solved using a saddle-point method, as I discuss in a previous answer of mine. One expands the field as $i \tilde{\sigma} = \Delta^2 + i \sigma$. Then we will obtain a partition function of the form
$$
\mathcal{Z} = e^{- N \mathcal{S}'[\Delta^2]} \int \mathcal{D}\sigma \, \exp \left[ \frac{1}{2} \int \frac{d^d p}{(2 \pi)^d} \left( \frac{\Pi(p)}{2} + \frac{1}{4 \lambda} \right) |\sigma(p)|^2 + O(1/\sqrt{N}) \right].
$$
We can drop corrections to the $N=\infty$ solution, and we have solved the theory in principle. One can show that the value of $\Delta^2$ which minimizes the action is given by
$$
m^2 + \frac{\Delta^2}{2\lambda} = \int \frac{d^dp}{(2 \pi)^d} \frac{1}{p^2 + \Delta^2},
$$
and I've also introduced the function
$$
\Pi(p) = \int \frac{d^d k}{(2\pi)^d} \frac{1}{(k^2 + \Delta^2)((k+p)^2 +\Delta^2)}.
$$
We now consider correlation functions of the fields. These can be comuputed, for example, by coupling a source to the field we are interested in in our original theory, carrying out the saddle-point expansion, and then taking variational derivatives with respect to the sources. For the fields $\phi_{\alpha}$ we find
$$
\langle \phi_{\alpha}(x) \phi_{\beta}(0) \rangle = \int \frac{d^dp}{(2 \pi)^d} \frac{\delta_{\alpha \beta} \, e^{i p \cdot x}}{p^2 + \Delta^2}.
$$
This implies that correlations of the $\phi$ fields decay exponentially unless $\Delta = 0$, in which case
$$
\langle \phi_{\alpha}(x) \phi_{\beta}(0) \rangle \sim \frac{\delta_{\alpha \beta}}{|x|^{d - 2}}.
$$
We can tune to $\Delta = 0$ by fine-tuning the mass term, $m_c^2 = \int \frac{d^dp}{(2 \pi)^d} \frac{1}{p^2}$. (We have a positive rather than negative $m^2$ because I defined $m^2$ with a different sign than is usual.) Of course, we should regulate this integral in the UV to get a finite value for $m_c$. The integral is IR divergent for $d \leq 2$; there is no gapless solution to the saddle-point equation in this case. In tuning $m_c$ to this value, we have tuned to the line between $G$ and $WF$ in the above picture.
We can conclude that, as a function of length scale $x$, the scaling dimension of the $\phi_{\alpha}$ fields does not change; it is equal to the free-field value of $D_{\phi} = (d-2)/2$ in both the $G$ and $WF$ CFTs.
But not all operators behave so trivially. Consider the O($N$) singlet operator, $\phi^2 \equiv \sum_{\alpha} \phi_{\alpha} \phi_{\alpha}$. By coupling this to a source field, one can show the identity
$$
\langle \sigma(x) \sigma(0) \rangle = 4 \lambda \delta^d(x) - \frac{4 \lambda^2}{N} \langle \phi^2(x) \phi^2(0) \rangle.
$$
So by studying the behavior of the $\sigma$ field using the above Gaussian theory, we can determine the scaling dimension of $\phi^2$. For $\Delta = 0$, it is not hard to show $\Pi(p) = F_d p^{d - 4}$ for an uninteresting dimensionless constant $F_d$, and we can read off the $\sigma$ propagator:
$$
G_{\sigma}(p) = \frac{2}{\Pi(p) + 1/(2 \lambda)} = \frac{2}{F_d p^{d - 4} + 1/(2\lambda)}.
$$
I will rewrite this to make its dependence on $\lambda$ more apparent:
$$
G_{\sigma}(p) = \frac{1}{p^{d - 4}} \frac{4 \lambda p^{d - 4}}{2 F_d \lambda p^{d - 4} + 1}.
$$
The point of this rewriting is to single out the dimensionless combination $\lambda p^{d - 4}$, which clearly controls the flow between the IR ($p \rightarrow 0$) and the UV ($p \rightarrow \infty$).
First consider the IR. Since we are assuming $d<4$, we find $G_{\sigma}(p) = 2p^{4 - d}/F_d$, so after a Fourier transform, we expect
$$
\langle \phi^2(x) \phi^2(0) \rangle \sim \frac{N/\lambda^2}{|x|^{4}}.
$$
We find that the IR scaling dimension is $D_{\phi^2} = 2$, which is not equal to twice the scaling dimension of $\phi_{\alpha}$ as in the free theory. Note that this precise power of $\lambda^2$ which appears on the right-hand side is required for the engineering and scaling dimensions of $\phi^2$ to match.
In contrast, in the UV one has
$$
G_{\sigma}(p) = 4 \lambda - 8 \lambda^2 F_d p^{d - 4} + \cdots
$$
After a Fourier transform, the first term on the right-hand side generates the delta function indicated above (with the correct factor of 4), and we find
$$
\langle \phi^2(x) \phi^2(0) \rangle \sim \frac{N}{|x|^{2(d - 2)}},
$$
indicating that the scaling dimension takes its free-field value $D_{\phi^2} = (d - 2) = 2 D_{\phi}$. So the UV of this theory is at the Gaussian fixed point. Note that the $\lambda$ dependence dropped out.
Of course, for intermediate observation scales, one needs to compute the entire function
$$
\int \frac{d^d p}{(2 \pi)^d} \frac{e^{i p \cdot x}}{p^{d - 4}} \frac{4 \lambda p^{d - 4}}{2 F_d \lambda p^{d - 4} + 1}
$$
to obtain how the correlator $\langle \phi^2(x) \phi^2(0) \rangle$ behaves as a function of $\lambda |x|^{4 - d}$, and it will not behave as a power-law (and therefore not describe a CFT) until you take the UV or IR scaling limits.
