I agree with Bailiff, you need to approach the problems as Bailiff cites " You basically have a perfectly normal completely untrained and understimulated dog."
Unfortunately, the situation you describe sounds as if your puppy is left to his own devices and many times that spells hardship for both the dog and his "family". My experience with shepherds would strongly suggest that 1: they are extremely smart dogs, 2: they need guidance and focused leadership while they are maturing, 3: they need consistency along with a guarantee of human interaction...the more the better. If your pooch becomes acclimated to the consistency of very little training and minimal human interaction coupled with a lack of stimulation, you will have a less than desirable outcome most likely.
I will be judgmental for a moment here and ask if the decision to get this pup was well thought out in the first place, too many folks in error believe a young dog is a self starter and requires little attention as well as training.
Your comment " I don't see it fair on the dog to be given away after he's become part of our family." shows where your heart is at and I applaud you for this. Perhaps, you are the only one in your family who feels this 1 year old pup is "part of the family". So, perhaps you might take it upon yourself to go the extra distance and do what is required even if you don't get any support from the rest of your family. The fact that you are taking steps to correct certain unacceptable behavior shows you care.
There are numerous resources available on the Internet to help guide you along but personally I believe the more time you spend with your pup the better off all will be. I know you say that you have limited time and are tired after working a full day but do everything in your power and ability to spend time with your dog, training and interacting with him. Simply put, you get back what you invest in your dog's upbringing. So, make the sacrifices and make your dog a top priority, you will be paid back many times over for your efforts. Maybe during your work day if it is possible to come home at lunch or whenever and simply work with your dog, you will see the benefits. And I would reserve weekends for your pup and have him glued to your side, playing and training as much as you can ( keep it fun when you sense you are making little progress ). Upbeat and positive with solid leadership will usually win the day.
A dog's desire for attention is puzzling at times but I have heard it said many times that "any attention is good attention to the dog". I take that as the negative attention your dog probably receives for chewing everything up he isn't supposed to, could be interpreted by your dog as simply "attention" and in his mind is better than no attention whatsoever.
A 1 year old shepherd has plenty of time left to become a wonderfully trained and compatible friend for life but the ball is in your court as you well know. It requires patience, effort and dedication for a while but once you get your dog on the right track you will be ever so glad you made the effort because you will have as I said previously a friend for life.
Your comment " I feel really bad since he doesn't get out much or any exercise and it's really not fair on him..." also shows you have a solid understanding of what he needs and is good for him. So, let your conscience guide you brightly and exercise him whenever you can...make the exception for him, he simply will be better for it. A five minute session of exercise is better than none and perhaps you can turn that 5 minute session into 10 minutes then 20 etc...and as many times a day as you can until you really sense your dog is tired out.
I wish you luck and I'm glad your pooch has you as a family member who cares......he deserves the best you can give him because if you do what you must today for him, he will fill your life with quality.