No and yes. Thats a newtonian reference frame problem, and is complicated because when the spit leaaves your mouth it is slowed or accelerated (depending on wether you gob forwards or backwards) by the wind of your passage. The point with relativity is that all reference frames are valid. to you at light speed everything seems normal, light recedes at the correct rate and all in the garden is lovely. You would be a bit lonely though, as you would effectively be in your own little universe. Time for you would have slowed down to zero (moving clocks run slow, remember) so to the rest of the universe you would have effectively ceased to exist. You would not notice this though. The direction of motion is irrelevant to the reference frame. It is only relevant if photons from outside are being considered (the falling lift problem) and at lightspeed the effective curvature of these would be assymptotic so they are irrelevant.
so there.