give me a second I'm just uploading my mechanism for the product formation...
(there are obviously many more possibilities, but I've shown the two most likely you get for each resonance structure)
@hansika yes, I feel that an electrophilic attack on an alkene has to be slower than simply attacking the alcohol to form a carbocation, because of two reasons:
(1) as you said, acid-base is faster, and an alkene is not extremely electron rich (2) in this case, by dehydrating the alcohol to get a carbocation we get a resonance stabilised carbocation while attacking either of the alkenes first gives us a much more unstable carbocation; so attacking the alcohol first makes much more sense to me (lower activation energy to first TS)