It seems to me there are two separate questions here. The first is how we know distant galaxies are moving away from us when we can't see them now - we can only see them as there were some time ago. The second question is why we say the distant galaxies are accelerating away from us.
To answer the first question you need to know a bit how how physicists work. It isn't enough for us to measure the recession velocities of distant galaxies. We want an explanation for those velocities, that is a theory to describe how the galaxies move. We get this theory by starting with Einstein's equations of general relativity, making a few assumptions (homogeneity and isotropy) and solving the result equations to get an equation called the FLRW metric. This equation tells us how the galaxies should be moving, and when we look in our telescopes we find our observations match the predictions.
So we have a theory that gives the correct predictions for what a galaxy $x$ billion light years away was doing $x$ billion light years ago, and since the theory was correct then we assume it's still correct today. That's why we feel confident in saying what distant galaxies are doing right now, even though we can't see them doing it.
The second question follows on from the first, because the acceleration of the universe expansion is also explained by the FLRW metric, sort of.
The FLRW metric contains some adjustable parameters. The obvious one is the density of matter in the universe, but there is also an adjustable parameter called the cosmological constant and a non-zero cosmological constant will cause an acceleration in the expansion rate. It's important to be clear that (at the moment) the cosmological constant is just a parameter and we have no theory to say what its value should be. We have to compare observations with the predictions of the FLRW metric and choose the value of the cosmological constant that matches. When we look at how far away distant galaxies are we find that for the FLRW metric to match experiment we need a non-zero cosmological constant, and that's how we know that the expansion is accelerating.
