It would be possible for beings with enormously long lives. If typically beings had lifetimes, say, of a billion years, you could easily go from one side to the other of the galaxy a thousand times at a tenth of the speed of light.
If you suppose also an ultra stable society - somehow they have dealt with all the issues we have - then - I could imagine it being possible. There might not be many beings in the galaxy in this civilization. The galaxy has a hundred billion stars, very approximately, more than ten times the number of humans on the Earth.
Perhaps it is immature to want to fill the galaxy with your species. And I think it is probably also potentially quite dangerous when those beings are short lived, and powerful able to handle enormously powerful technology and biotechnology, such as self replicating machines.
I wouldn't want to seed our nearby stars, say up to 100 light years with present day humans with that kind of technology - any moment then some of them might come streaming in out of the night sky with vastly superior technology that they invented while away from Earth with some obscure demands on Earth based on cultural ideas we no longer recognize or understand. And they could also have modified their bodies and ways of thinking, or created new machines or biological beings or cyborgs, and even as close as 100 light years away - they could be in populations of billions already, trillions even if they are fast breeding, or rapidly self replicating, before there is any chance of seeing any effects with speed of light limits on observation of their star systems from Earth.
And also I think not at all clear that they would be bound to be a good influence on the galaxy either. Do we want to be the monsters that other peaceful ETs in the galaxy fear?
So - I think that ETs and ourselves if we have any sense will be cautious and careful until we are mature enough to see a way forward through ethical dilemmas like that.
And when we can, with billion year lives most likely by then, we might be altogether happy with a few hundred billion humans in total in the entire galaxy - especially if we find it has other inhabitants in it also. Say average of one or two humans per star, but of course concentrated in some places, or maybe traveling or exploring the galaxy?
If so that might be why we haven't seen ETs yet, because though they are there and are maybe quite widespread through the galaxy, we are due a visit some time in the next million years or so, which to them seems a short time, and nearest ones, maybe a small settlement a few tens of light years awaya. I mean is one scenario.