I worked as a dog trainer for 8 years, going into people's houses, meeting new dogs, plenty with fear and aggression problems. I agree that you should not rush over to pet them, let them come to you. But if you stand there looking away from him without saying hello, many dogs are going to think there is something wrong, bc they are used to being greeted by friendly people who call out to them in an inviting, friendly voice and attempt to make contact. Watch two dogs meet on the street. If they are friendly they look towards each other (without making a hard glare) and approach with tail wagging loosely and loose body language in general. As they get closer, they circle behind each other, avoiding direct nose to nose and eye to eye contract.
Try to do the human equivalent of this. Smile, turn your body slightly to the side in a relaxed posture and call him. He should reciprocate with loose muscles and a slow easy wag to his tail. When he comes over, pet him under the head (never over the top til he trusts you) and talk nicely to him. You don't need to establish your dominance within 2 minutes of meeting a dog. Establish your good intentions and build a relationship built on trust, not fear.
And, if you have to be super careful about avoiding eye contact or if he is stiff, afraid, growls, etc., I wouldn't adopt him anyway. You want a dog with good social skills so adopt a pet that is nice, social, open and friendly. Let the macho people who want a protection dog adopt the distrusting GSDs of this world. It wouldn't be unusual for him to feel a little nervous, but a dog inclined to be a good pet will resolve this nervousness with submission, appeasement, calming signals, etc. not avoidance or aggression or hyper-activity.
When you introduce him to Mary I would make sure that they like each other. If she is very submissive you certainly don't want to bring a bully into the house- that won't be any fun for her. Better to pick the right dog than to try to change the wrong one.