Short answer? We cannot save Earth
Saving Earth from it? I don't think theres a possible way of doing that.
The reason that stars swell in size after exhausting their hydrogen cores is that the stars begin fusing helium, according to the University of Michigan. The helium releases more energy during fusion than the hydrogen releases during the process. The amount of energy produced by the helium is greater than that needed to stave off gravitational collapse, which causes the star to swell greatly in size.
Trying to slow down sun advance
Refuel the hydrogen core won't do nothing, even thought our sun still has a lot of it that would make nothing, let's not say that amount of hydrogen would be a lot.
Reducing the size of the sun? Well, I don't know if this is possible or not, if we can even have a machine that can do such task. Also humans can get near the sun, so we would need a machine to operate from distance or a machine controlled by robots that won't melt being near the sun.
Also, what would this machine do? If a machine like that is possible.
Of course I'm using the perspective view of a 21th century man, we really don't know what we may develop in the future. But in terms of what we know right know there's no possible way of slowing down or avoiding our star to devour us.
But, that would take time, around 1.75 billions years according to scientists, so...
Traveling to another planet
If you want to save flora and fauna people has always come with the expectation that we can find another habitable planet where we can land our already know flora and fauna. But I always wonder if that would be possible due to the fact that would be like introducing a new specie to an environment, and that has not always had the best results, but it may be possible if the new planet has the condition the flora and fauna needs to survive.
Eventhought there are a lot of post about moving the planet we need a tremendus force to achieve that. We are not lifting a plane, we are moving a dense rock that wieghts around 5,972 × 10^24 kg. If we can achieve a source of energy capable of doing such task then we would need to figure out how to create a machine capable of moving earth.
So, in my opinion, we have around 1.75 billions years (if we don't destroy ourselves before) to figure a way out of this planet, because I think is the most viable method. Slowing the sun seems impossible and moving earth seems out of reach.