Sorry this has taken me so long; It's not the generating of the concepts, but the typing them out.
I assume that the orbit has become ever more elliptical, and the perigee is now close enough to the primary to be developing friction? A steady, slow spiral wouldn't give you a few years to solve the problem, but a few centuries. Much preferable. But now we get long, very cold winters, and short but drastically hot summers.
So, how do your ships' drives work, and how much damage am I allowed to do? Jogging a planet ,even the tiny bits to recircularise an orbit, is going to take immense quantities of energy; but, so does accelerating a couple of million tonnes up to near light speed. I was considering using the magnetic field generator used for compacting interstellar hydrogen for a bussard ram jet, and eddy currents in the planet's core, but even that, which can only hope to provide millimetres per second squared (and the ship will lose almost all of its momentum, and be useless for days). Even that will increase vulcanism and planet quakes; some of my other tricks (like colliding it with extremely accurately aimed comets, which would obviously vaporise in the atmosphere, but their momentum would be conserved. As would their cargo of volatiles; you can expect some serious sea level rise as it condenses out as rain.
If you had my gravitic drive, the mechanism is dead simple; but the difference in mass makes moving a planet with ships a bit equivalent to water skater beetles towing an ocean liner. There's nothing much stopping them, but the process is going to take a while. Probably more than the years before surface conditions become life supporting.
So, we set up manufacturing tens of thousands of huge robot ships. Each one swoops past the planet on a calculated trajectory, close enough for some spectacular tidal effects, and gravity to transfer a fair amount of momentum. Then it redirects itself back out to the gas giant, where orbital stations are preparing more fusion fuel and reaction mass, to accelerate up and swoop again. As far as possible everything automated, with humans trying to clean up the inevitable crises.
OK, one slightly better in effectiveness, if slow. We build a really big fusion jet – big as North America, at least, probably as big as Earth – and find a small gas giant, something like Neptune or Uranus (not a gas dwarf by any means; still many times the mass of our target planet) Now we plunge our motor into the gas planet and light it up, at a throughput of several hundred hydrogen bombs a minute (gotta be tough). The plume of vapour jetted into space shifts the planet's orbit into something more elliptical, until finally its perigee is within the orbit of the potentially habitable planet. Lots of careful manoeuvring (of a mass considerably greater than the universe was supposed to be until the eighteenth century) and it's in just the right place that its gravitational field pulls the planet in the direction you planned, and again as it rounds the sun, and goes skooting off into the cold and dark for fifty years or so, when it becomes available again for more Velicovsky cosmic billiards.
Oh, stupidly massive tides, earthquakes, hurricanes to dwarf anything ever seen in our atmosphere; I get to break things. But probably not the entire environment.
Trouble is, building that jet is going to take a century, (and joggling the big planet's orbit decades, at least). By the time we're ready to act, it's too late. So, how about this? We build the thing in orbit in the solar system, when the first probe sends back information about a potentially habitable planet, and put the fleet in orbit around it.(It'll mass as much as a fair-sized moon) Then we feed it comets, and it accelerates, towing the space stations and terraforming vessels that can remain fully fuelled the whole trip. They can even use solar cells to get energy from the yellow hot fusion engine, and greenhouses can grow food. The idea being, evidently, that they cut loose in the destination system; but when they are a decade or two out, they spot the orbital decay of their potential new home. For cosmic engineering you need large tools, so they allow their tug to be captured by a planet in the system, and start doing their sums…