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You wrote:

Just to give the background...I have the vision of an ancient race who's star saw it's end. They relocated to a star moving significantly slower than the majority of those in the galaxy, and as such the galaxy around them aged quickly while they saw a relatively short time span pass.

How feasible is this?

Not very. It does depend on a few things, though.

Velocity time dilation

According to special relativity, an object moving relative to another measures time differently; this phenomenon is known as time dilation. An observer in motion measures a time interval $$\Delta t'=\frac{\Delta t}{\sqrt{1-v^2/c^2}}=\Delta t\gamma$$ where $\gamma$ is known as the Lorentz factor and $\Delta t$ is the interval measured by a stationary observer. We should be able to compute $\gamma$ for any star in a galaxy.

Stars don't orbit like a rigid body rotates (e.g. where their tangential velocity is proportional to their orbital radius). Similarly, they don't have Keplerian orbits. In reality, they follow rotation curves - plots of velocity as a function of radius - that are complicated, thanks to both dark matter and the fact that the galaxy isn't a point mass.

I decided to look at some real data, so I turned to galkin (more information here), which has data from a large number of observations of the Milky Way. In particular, it generates rotation curves from data from gas clouds, masers, and stars.

I took the speeds of the stars tangent to the galactic center as computed by galkin. I first plotted a rotation curve, which is fairly flat beyond $r=5\text{ kpc}$.:

I then calculated $\gamma$, the Lorentz factor, relative to a stationary observer at infinity: $$\gamma=\frac{1}{\sqrt{1-v^2/c^2}}$$ where $v$ is the star's tangential velocity. As expected, $\gamma$ is close to $1$ for all $r$; there is little time dilation due to the motion of stars.

Gravitational time dilation

Time also flows at different speeds for an observer in a gravitational field. The galaxy's gravitational field isn't large, and so we can apply something called the weak field limit: $$\Delta t'\approx\Delta t\sqrt{1+\frac{2\Delta\phi}{c^2}}$$ where $\Delta\phi$ is the difference in potential between two observers. To compute the potential, I used a model by Flynn et al. 1996. It reproduces the observed rotation curve very well, using a three-component model:

By setting an observer at infinity, I let $\Delta\phi=\phi$, and thereby calculated the proportionality constant at all $r$. It turns out to be quite close to $1$, meaning that there is very little gravitational time dilation.