Short answer is: yes.
Longer answer is: correcting for the time dilation effects of Earth moving around the Sun's gravitational potential is actually relatively standard in almost all branches of astronomy. To the point where running that correction is a sentence in a paper (sometimes less), and is probably why you had trouble Googling for it.
(I'll caveat all this by saying I'm mostly familiar with exoplanet transit and RV timing issues, but they should be the same as what the pulsar folks have to deal with).
As some background, the base time-keeping system used around the world is International Atomic Time (TAI), which is a weighted average of over 300 atomic clocks determined by the International Bureau of Weights and Measures outside of Paris. Importantly, TAI is strictly continuous: there are no leap seconds added. This is important if you care about sub-second timing precision.
What we use as normal "clock" time is Coordinated Universal Time (UTC), which is TAI with leap seconds subtracted off. Those leap seconds are present to deal with the fact that 86,400 SI seconds are 1 to 3 milliseconds less than one mean Solar day, and so ensure that our clock time is linked to the position of the Sun. The most recent leap second was added just this past New Year's, making UTC = TAI - 37 seconds.
Even further down the time-keeping rabbit hole is Barycentric Dynamic Time (TDB), which accounts for the variable relativistic time dilation over the course of a year that you asked about. TDB has a fixed offset from TAI of 32.184 seconds due to how the zero-points of the the two systems were defined, and otherwise stays within 1.6 milliseconds of TAI - depending upon where Earth is in its orbit.
Effectively all precise times reported by astronomers these days are the barycentric Julian date in the Barycentric Dynamic Time system (BJD_TDB). This is the Julian date an event would appear to happen for an observer located at the Solar System's barycenter using TDB as their timekeeping system. Note that the fact this is at the SS barycenter does matter, since observations on Earth will see similar events up to ~16 minutes apart over the course of the year due to light-travel time delay (Roemer Delay, for the aficionados) across the Earth's orbit.
So yeah, this all has to be accounted for all the time. As I said, these days the transformation is standard enough that you usually just list a time as "BJD_TDB" and don't have to explicitly discuss the transformation.
For more reading about astronomical timekeeping, see Eastman et al. (2010).
PS - In case you're wondering why Barycentric Dynamic Time is abbreviated TDB and Coordinated Universal Time is UTC, it's because we all use the French abbreviations.