You poor love, you are not a bad person. A bad person wouldn't feel as you do, wouldn't be asking for advice and their dog wouldn't still be in the family home.
Re a dog sitter - maybe ask your local vet and pet shops if they know anyone who could help for a small fee? Even a neighbour? This daft beggar has dog sat for both a neighbour and when my pet shop suggested me to a young lady with an escape artist Greyhound with separation anxiety. I jumped at both chances, doing so for the neighbour for love of the dog, the stranger for a fiver here and there until I got to know the lady and said blow it, keep your money in your purse hun, its a pleasure to have your pooch here. That was a few years ago and the lady and I are now and have been for some years firm friends.
Neutering doesn't ALWAYS bring results to the kind of problem you are experiencing, but in my experience does so more often than not. Local training classes can be a blessing too (and are a great way of gaining advice, friends and perhaps that dog sitter). My own town's trainer charges £50 for an 8 week course, one evening a week, and I am near a major University city so I'd guess this is on the middle to upper scale of charges - dependant on where you live your local classes may be cheaper than my area.
I'd also say that if you can cope with kids, hey, you can cope with anything... don't put yourself down! DD1 was the first baby that I had ever held although I was 30 when she was born - and she and I are still in one piece! (Though I've been tempted to send the bugger back a few times over the past 14 years!!!)
Remember, your dog is still a baby... well, a toddler in dog years... and it is now you need to set the ground rules just as you taught your DC not to touch the cooker/bite their sibling/s etc. Its early days and yes, as a new owner it must be daunting to you.
You don't say if you have a partner - its no big deal in the world of dog owning if not, you can still cope, I do as a lone mum - but if you do perhaps you could ask him to support you with the strong tone etc?
All young dogs mouth, don't stress, though of course there is a difference between this and snapping or biting. One tip if he mouths is to shriek OUCH in a fairly high pitched tone, just as his baby brothers and sisters would squeak if he inadvertantly hurt them when playing. He will soon learn that this noise means he has overstepped the mark.
As for the barking when you go out, thats seperation anxiety and best nipped in the bud if possible. Leave a radio on in the room when you go out and if you are not going to be gone for long pop him in a crate where he will feel safe - dogs like their own "hidey-hole". Again, introduce him to a crate by persuasion, the door open, cuddly blanket and toy inside (though make sure he can't hurt himself by eating the eyes of a toy teddy or the like). He will soon consider that "his" place and retire to it when he wants to, so making popping him in when YOU want him there easier.
The other way to deal with this is a well known one to rescue workers and fosterers -its NOT to make a fuss of him before you go out or immediately after you return. And... don't laugh, though neighbours must have peed themselves at me with my foster dogs, but go through the motions of going out but return in 5 mins, gradually extending the time taken to return. Pick up your keys, put on your coat, bang the door shut, give it a mo on the doorstep, then re-enter, totally ignoring pooch. He will eventually realise that you aren't abandoning him.
On a lighter note, this method can make an even bigger fool of you though. A friend and fellow fosterer did this recently with her foster Lurcher, who cries when she leaves although her own SIX smaller dogs are quiet as lambs. So, Amanda puts her coat on, picks up her keys, jangles them, slams the door, creeps round to the side of the house and stands listening (while her neighbours are looking out of their windows thinking she has lost the plot).
Nothing.
Complete silence.
She thinks she's cracked it!
She walks back in and into the kitchen... and discovers why its so quiet.
The bread rolls which she had bought for her husband's lunch have been removed from the kitchen surface and were on the floor, merrily being munched at by all seven dogs... but there was only ONE which was tall enough to reach them!
Btw, where does pooch sleep? Just thinking, if he sleeps on a human's bed (although again, I'm guilty, my GSD and Labrador sleep on mine), yours might be getting a reinforced message that he is top dog and you are there to do his bidding. If this is the case, again, a crate or popping him in the another room (although he might complain for a couple of nights) might help disabuse him of this notion.
Finally, sorry for jabbering on, but PLEASE don't be so hard on yourself. If you could only see some of my posts on my rescue's forums about a couple of families who HAVE given up too soon without caring and without trying, you'd realise why I (who can be REALLY stroppy about owners who don't make an effort!), have every respect for you in the world and will do my utmost to help if I can.