Dehydration of alcohols (in case of secondary and tertiary) proceeds through a carbocation intermediate, as you already have mentioned. You can refer this link in Master Organic Chemistry for the full mechanism.
In this case, the formation of carbocation is the slowest step, hence the rate determining step. So the stability of carbocation must be considered in the mechanism. After the formation of the initial carbocation, it can rearrange either by methyl shift or hydride shift. I could spot two reasons why hydride shift will be preferred -
- Hydride has more migratory aptitude than methyl
- After methyl shift, the new carbocation will have 6 alpha hydrogen for hyperconjugation. By hydride shift it will have 7. The latter is hence more stable and preferred.
After this step, when the new double bond is formed, it can give either A or C as you said. (A) will have more hyperconjugative structures than (C) so (A) is major.