Well I think you might still experience time as a short trip but everything else would age as if you had completed the physical distance?
For some reason I never see this discussed when people talk about FTL travel, maybe I'm wrong?
This is how the twin paradox gets solved. The twin that leaves earth sees the trip as if it was shorter in both directions, so from their perspective it makes sense that they aged less than the twin that remained on earth.
Also, if you had infinite energy and you could travel at the speed of light you wouldn't feel any movement or time passing during your trip, it would feel like instantly teleporting from one place to another. Photons wouldn't feel their existence if they could. From their perspective they are produced in one place and instantly absorbed in another place.