As others have said it is possible that the bike is just too long for you.

However, it may just need some adjustments. It sounds like regardless of bike size the shop set you up in too aggressive a position to start with (or didn't set you up at all).

If there is still steerer tube left you can move the stem up and drop the spacers below it. You can also change to a shorter stem, one with a higher rise (angle), or both. Switching the bars from standard "male" bars to what they call "short and shallow" (or just short) may help as well (brings your reach back some and makes getting in the drops and up on the hoods easier). Would also be a pretty good idea to measure your shoulder width (collarbone I believe) and compare that to the handlebars, they may well be too wide which will further cause you to feel like you are laying down (or falling flat on your face) and contribute to neck and upper back pain.

I'm 4'11" and my main rides are not WSD. It has little to do with your height and much more to do with body proportions. (as you are learning)

I'd go back to your shop. Explain your pain and see what they are willing to do to see if this bike will indeed work for you before concluding it doesn't fit at all (my bikes all have custom picked bars, stems, seats, and seatposts to dial in fit...it's not unusual). Hopefully those few minor changes will help you get the bike dialed in without having to go through the headache of trying to get the shop to exchange it (and if the shop is good they'll be willing to spend some time and work out some deals working with you on this).