Relativity has almost nothing to do with the fact that signals have a 38 hour there and back transit time. This time period is because the spacecraft is so far away, roughly 19 light-hours in distance, that radio signals take 19 hours to reach it and the reply takes 19 hours to return.
Technically the return time will, on average, be marginally longer than the out time but that's more because the spacecraft is slightly further away from Earth by then than any other effect. How much this matters compared to Earth's orbit is another matter. Earth's orbital speed is 30 km/s, roughly twice that of Voyager 1, this is in a roughly circular orbit therefore roughly half the time Earth's orbit will be increasing the distance and the other half it will be reducing the distance but there will only be short periods when the maximum relative differences in velocity come into play.