Personally, I find that CoefficientRules/FromCoefficientRules is by far the most robust way to manipulate multivariate polynomials without having to worry about arcane and ad-hoc parsing tricks. For the problem of finding terms of order $\leq1$, we can simply use a Select to filter the total exponent of the monomials:

Arules = CoefficientRules[A]
(* {{1, 1, 1, 1, 0, 0} -> -1, {1, 0, 1, 0, 1, 1} -> 
     1, {1, 0, 0, 0, 0, 0} -> 1, {0, 1, 1, 0, 1, 1} -> 
     1, {0, 1, 0, 0, 0, 0} -> 1, {0, 0, 1, 1, 0, 0} -> 
     1, {0, 0, 0, 0, 0, 0} -> -1} *)
FromCoefficientRules[Select[Arules, Total[Keys[#]] <= 1 &], Variables[A]]
(* -1 + a + b *)