Just to echo what others have said - one month is such early days.

Dexter has been with us for ten months and he still follows me inside the house from room to room.

I am trying to teach him the 'stay' command to stop this happening and sometimes it works, but it's still a work in progress.

Try and relax because the more you relax, the more your dog will relax too

Yup! This normal behaviour for GSDs. Be flattered.

What a sensible owner you are, and you are perfectly correct, in order for a dog to become a competent and confident individual, alone training is a must.

I start alone training from 7 weeks old, and when I had a rescue I started it from Week 1 he was with me.

I would suggest starting as you would a puppy, slowly to ensure that the training is progressive and incremental to avoid increasing stress etc.

So I would start by being able to distance yourself from the dog whilst you are in the same room. I fhe can be "alone" without touching you or you speaking to him comfortably only THEN can you start doing some departures eg for a nano second and then returning.

It is really important to minimise the contrast of ALONE v TOGETHER by detaching yourself a little whilst you are there and when training departures that both these and arrivals are NOT times of high anxiety or excitement.

If you need any more info, I can post you some helpful info.