Hi there and

!
First and foremost, don't ever be ashamed to go to they gym!!! Believe it or not, most people are just too into what they're doing to notice anyone else. I just got back from the gym and I couldn't tell you if there was someone there who was purple with green polka dots.
If you are intimidated or lost as to what exercises to do, maybe you should sign up for a session with a trainer to show you the ropes? I go to the YMCA and upon signing up you get a free session with a trainer. Perhaps your gym has something similar?
As for your time limit, those are some pretty rough shifts, but if you want this you can make the time. You say you have a 4 on 3 off schedule? Can you commit to working out on your off days? Or maybe, if you need a day off 2 of your off days and 1-2 of your work days? Also, as it sounds like you're pretty serious about getting healthy, maybe you can get your DF on board... maybe he can cook his own dinner a night or two a week so that you can focus on you (that is what's important, here

)
And congrats on your engagement!