There are many reasons this is a bad idea
Distance of Earth at 372 day rotation
First we have to find the radius of our new 372 day orbit. By Kepler's third law, orbital period is $$T = 2\pi\sqrt{\frac{a^3}{\mu}}$$ where $\mu$ is the standard gravitational parameter of the sun ($1.327\times10^{20} \text{ m}^3/\text{s}^2$), $a$ is the semi-major axis of the Earth's orbit ($1.496\times10^{11} \text{ m}$, at least for now), and $T$ is in seconds. If we plug in the numbers above we get $T = 31560349$ in seconds, or 365.28 days. Pretty good!
Lets set $T = 32140800$ for 372 days and solve for $a$. We get $1.514\times10^{11}$ meters, a marginal increase.
Temperature drop due to this distance
Now lets find out how hot this makes our planet. The effective temperature of a planet at a certain distance from the sun is given by $$T = \left(\frac{L(1-A)}{16\pi\sigma a^2}\right)^{1/4}$$ where $L$ is the luminosity of the sun ($3.828\times10^{26} \text{W}$), $a$ is the distance to the planet, $A$ is the albedo of the planet (0.3), and $\sigma$ is the Stefan-Boltzman constant, $5.67\times10^{-8} \text{ J /(K}^{4}\text{m}^2\text{s}$).
If we plug in the characteristics of Earth, we get 254.59 K. Not exactly super-accurate, mostly because there already is a good greenhouse effect keeping some of that heat in. However, if we calculate the temperature at our now longer orbital distance, we get 253.07 K. This at least gives us an estimate of the temperature delta we are looking at: if we can move the Earth out to this longer orbit then we might drop global temperatures by about 1.5 K. Since warming vs. pre-industrial is already around the 1 K range, this isn't that effective in the first place.
Energy to move the Earth's orbit
But wait! We still have to have spent the energy to move the Earth in the first place! Specific orbital energy can be calculated by $$E = \frac{\mu}{2a}$$ with symbols as above and ignoring the sign. For our current orbit we then have a specific energy of $4.435\times10^{8}\text{ m}^2\text{/s}^2$. For the farther orbit we have $4.382\times10^{8}$ for a difference of $5.273\times10^{6}\text{ m}^2\text{/s}^2$. Last we multiply this number by the mass to the Earth ($5.972\times10^{24} \text{ kg}$) to find that we need $3.15\times10^{31} \text{ J}$ to move the Earth to this new orbit.
Using the world's greatest internet page we find that this is equal to the sun's daily energy output, or about a million years worth of Solar energy on the surface of the Earth. This is going to be tough.
Even if we did move the Earth...
Now, assuming that we move the Earth, there will be some waste heat in this process transferred to the Earth. Lets make the very unreasonably low assumption that only 1% of the energy expended to move the Earth is transferred to the Earth as waste heat. That means of the $3.15\times10^{31} \text{ J}$ needed to move the Earth, $3.15\times10^{29} \text{ J}$ will be delivered to Earth's atmosphere and hydrosphere. This much energy is enough to raise the oceans (mass = $1.4\times10^{21}\text{ kg}$; specific heat $3850 \text{ J/kg}\cdot\text{K}$) by about 6 million degrees Kelvin.
Put another way, the waste heat from moving the earth over the course of 11 years is enough to raise the temperature of the oceans by 1 degree K every minute for the entire 11 years. That is a heavy price to pay for 1.5 K of cooling.
Conclusion
This is not a thing that will work.