How to implement gain and offset in a single opamp?
Let's look at a simplified diagram:



simulate this circuit &ndash; Schematic created using CircuitLab
Let's set \$R_f=68\:\text{k}\Omega\$.
You know that \$1+\frac{R_f}{R_{th}}=3.5\$, which is your desired gain. So it follows that \$R_{th}=27.2\:\text{k}\Omega\$. If the input is \$4.5\:\text{V}\$ then the output is \$3.0\:\text{V}\$ and \$V_{th}=5.1\:\text{V}\$. Similarly, if the input is \$6.5\:\text{V}\$ then the output is \$10.0\:\text{V}\$ and \$V_{th}=5.1\:\text{V}\$, again. So we know what \$V_{th}\$ is!
To achieve that, we can use either the \$+12\:\text{V}\$ and \$0\:\text{V}\$ rails or else the \$+12\:\text{V}\$ and \$+5\:\text{V}\$ rails. In my comment I chose the latter pair. But in reality it's better to choose the former pair.
Let's expand the above diagram now:



simulate this circuit
We know we want \$5.1\:\text{V}\$ from the \$R_1\$ and \$R_2\$ divider. I've set \$R_1=27\:\text{k}\Omega\$, arbitrarily. So I can work out that \$R_2\approx 19.957 \:\text{k}\Omega\$. Round that up to \$R_2= 20 \:\text{k}\Omega\$. From this, I can work out that \$R_3\approx 15.711\:\text{k}\Omega\$. If we round that down to \$R_3\approx 15\:\text{k}\Omega\$, the gain will be just a bit high. But perhaps acceptable.
So the final diagram is:



simulate this circuit
That's not all that complicated. The output will be about \$2.94\:\text{V}\$ when the input is \$4.5\:\text{V}\$ and about \$10.08\:\text{V}\$ when the input is \$6.5\:\text{V}\$. As I said, the gain will be just a bit high. But this is pretty close. And feel free to adjust resistor values as you see fit to make this better.
Keep in mind that resistors have tolerances. So while my calculations above may seem close, you really need to apply worst-case scenarios here to see just how bad it may actually be given any particular resistor accuracy estimate.
For example, assuming exact values you'd anticipate that the node where the three resistors meet would have exactly \$V_{min}=4.8433735\:\text{V}\$ and \$V_{max}=5.7108434\:\text{V}\$ with those resistor values and given your input voltage range. However, 1% resistors may yield:
$$\begin{align*}
4.8092322\:\text{V}\le V_{min}&amp;\le 4.8778686\:\text{V}\\\\
5.6737453\:\text{V}\le V_{max}&amp;\le 5.7476844\:\text{V}
\end{align*}$$
With similar shifts in the resulting output offset and gain.
Just keep all this in mind. It is annoyingly difficult to get precision values at the output, like this.