Is the barycenter of the solar system dragging me along through the galaxy?
What I don't understand is that once I leave the atmosphere of earth and reach the vacuum of deep space, where the earth's gravity is no longer keeping me in orbit
(minor quibble: you're still in an orbit, just not a closed orbit. This will be a hyperbolic trajectory. Your motion through space will still be curved to some degree by Earth's gravity, just not enough to pull you back again)
As Ingolifs pointed out, once you've reach terrestrial escape velocity, you'll end up in a heliocentric orbit that's probably quite similar to the Earth's own orbit about the Sun (depending on how fast you're going, of course). The Sun's escape velocity is somewhat higher than Earth's, so you'll need to be going at least another 12.3km/s faster (which is solar escape velocity minus Earth's orbital velocity) in order to escape the Sun's gravititational pull and travel into interstellar space. And then you'll be orbiting the galactic centre. If you reach galactic escape velocity, you'll be presumably orbiting the barycentre of the Laniakea supercluster (which of course has its own escape velocity). Such is life with a force with effectively infinite range.
why do I not behave like a tennis ball that was thrown too high in a convertible?
On the assumption that you're not driving your convertible in a vacuum chamber (you'd be hard pressed to find one big enough... maybe the moon landing conspiracists can point you in the direction of one?) when you throw that ball up its forward velocity will initially match that of the car but drag will slow it down and so it will drift backwards with respect to the car. In space of course there's not really enough stuff to provide much drag under normal circumstances, so you'll just keep on travelling with the velocity you had when you were "thrown". If you escape Earth's sphere of influence, you'll initially enter into an Earthlike heliocentric orbit, because of the velocity you had before you escaped.