There will be some variation due to the eccentricity, but as that is small, so are the speed variations (which are periodic, as you say)
The earth is always accelerating toward the sun, and vice-versa. The magnitude of those accelerations are different by the ratio of the masses. If I've done my calculation right, the earth is accelerating at about 6 mm/s2 toward the sun. The sun's acceleration will be almost six orders of magnitude smaller, and I'm ignoring the effects of other planets (which are cumulative, since a is a vector)