Yep. Space is expanding everywhere at once, but the effect is minuscule at the scales we're used to. And even at galactic scales the "speed" of expansion might seem like a lot to us, but it still isn't enough to overcome the motion of objects. I looked up some rough numbers to give you an idea:

The rate of expansion of space is 73 km/s/Mpc. So for every 3.26 million light-years between you and a distant galaxy, the space between you and that galaxy is expanding by 73 kilometers per second.

Andromeda's blue shift indicates it's headed towards us at 110 km/s. And in my non-expert head I'm thinking that blueshifted light must have already been redshifted by the millions of years traveling through space to reach us. So the galaxy's speed through space towards us when the light was emitted was considerably higher.

Andromeda is 2.5 Million light-years away, btw. So the cumulative distance of space between here and there is expanding at something like 73 km/s/Mpc * 2.5 Mly * 1Mpc/3.26Mly = 57 km/s.

But when talking about relativistic distances and speeds, basic terms regarding time and location don't always make sense.

Milky way and Andromeda are close enough that expansion is too small to overpower gravity

Our local group is racing toward The Great Attractor but will never reach it as expansion is pulling it away faster than we're falling toward it

Yes, but not because of universal expansion really, they're just headed in a direction that is going to intersect at some point, likely combining the two galaxies together. This'll be so far in the future we'll all be long dead though, on top of it being unlikely that anything is going to even come close to our solar system