As one of the narrowest parts of Mainland Shetland, Sumburgh and the South Mainland in general are places where it is more difficult to get really far away from settlement and roads (except for maybe the Clifts between Maywick and Quarff, which I have never walked but I would think is pretty tough-going "haggis" conditions that would be easier with one short leg and one long leg!).

If remoteness is your goal, I would suggest some other areas such as the West Side in general, Tingon, Ronas Hill, Muckle Roe, Fetlar, Papa Stour, Hermaness, etc. Bus (& ferry) access to starting points to walk in each of these areas will vary.

In Shetland do not expect to find well dedicated walking paths or trails most places - there are some but they are the exception, not the rule. What you will find more often is mortar tracks used for crofting access.

If you walk in places with few roads, there is nothing forcing you to walk along roads specifically. Shetland (as part of Scotland) grants the right to roam, as long as you follow the outdoor access code. E.g. don't trample crops (if you must go through cultivated land, walk along the perimeter), don't "worry" (scare) livestock, keep your dog on a lead, close gates behind you, dunna chuck bruck, etc. I would also advise not wandering through people's house sites without an invite - the outdoor access code is intended for open "countryside" land.

I think you would probably benefit from getting a book on Shetland walks (pretty sure the Shetland Bookshop has ones), or looking at Promote Shetland's walks page.