As the Sun dies, does the Earth's orbit change? As the Sun nears the end of its life and expands, would the Earths orbit get bigger as it moves out or would it stay in the same orbital plane and not move out?  Therefore in the far off future, we'd have 512 days instead of 365 days?

As the Moon is moving out at a distance of 3.8 cms a year (ref: http://curious.astro.cornell.edu/about-us/37-our-solar-system/the-moon/the-moon-and-the-earth/111-is-the-moon-moving-away-from-the-earth-when-was-this-discovered-intermediate), are we moving out or in towards the Sun or is this not possible to work out because there is nothing to measure against?
The answer is yes. As the Sun ages, it will become a red giant and the mass loss rate from its surface will increase. This effect will increase (dramatically) further when the Sun enters the asymptotic giant branch phase, where thermal pulsations drives a cool wind that may carry away a millionth of a solar mass per year, eventually leaving a burned-out core in the form of a white dwarf with about half a solar mass.

At any point in this evolution we can model the evolution of the Earth's orbit using some simple approximations - that the wind from the Sun escapes to infinity, that a negligible proportion is actually accreted by the Earth and nor does it exert a torque,  that the mass loss takes place on a timescale much longer than the Earth's orbit and that the mass of the Earth $m$ is always much less than the time-dependent mass of the Sun $M(t)$.

In which case we consider the orbital angular momentum of the Earth:
$$ m a \omega^2 \simeq G\frac{M m}{a^2},$$
where $a$ is the semi major axis. So the angular momentum $J = m a^2 \omega$ is given by
$$ J^2 = m^2 a^4 \frac{G M m}{m a^3} \propto M a$$

As the angular momentum of the Earth's orbit is conserved, the $M(t) a(t)$ is constant and as the Sun loses mass, the semi major axis increases by the same factor.

Coming to the specifics - when the Sun is a half solar mass white dwarf, the semi-major axis will be 2 au (assuming the giant Sun did not quite engulf it - it will be a close-run thing) and Kepler's third law $(P^2 \propto a^3/M)$ can be used to estimate an orbital period of 4 years.

The tidal effects of the Sun on the Earth's orbit are quite negligible compared with these mass loss effects.