I have never heard the word 'dobber' used as a sectarian insult, and I've lived here all my life!
In my experience it has always been used to liken someone to a penis, regardless of what team they choose to follow.
As for OP's question - although I grew up in the southside I'd echo coffeeking's suggestions for the north. The landscapes there offer more hint of the not-so-distant highlands.
South is alright, but you seek quietish, with gardens - Giffnock/Merrylee perhaps, Clarkston/Muirend maybe, Cathcart/King's Park even. Not sure about 'pubs' there, in the way that I understand English ones to be, but still.
I'm sure another STW user did some asking on here before making a move to Glasgow. Anyone recall who that was?
Don't know if you get a recon trip before you move, but I'm sure you'd get a warm welcome and a couple of volunteer tour guides.
best of luck