a. Instead of running straight ahead, ~
b. Instead of straight running ahead, ~
Q1) Are both of them grammatically correct?
Q2) If so, do they have the same meaning?
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A is grammatically correct and in frequent use. Here "straight" forms a phrase with "ahead." Where is he running? Straight ahead.
I don't know if B is technically grammatically correct but I have a hard time imagining a native English speaker ever saying such a thing. Straight in this case would be an adjective describing the type of running. There may be some context where "straight running" as opposed to "crooked running" is used, but it's not normal, proper English.