= Solution
<protonation> of the <alcohol> converts its OH group into a <water> <leaving group>. Departure of <water> initially gives the secondary <carbocation> $\mathrm{CH_3-CH^+-CH(CH_3)_2}$. A \b[<1,2-hydride shift>] from the neighboring <methine group> transfers its C–H bonding pair to the electron-deficient <carbon>:
$$
\mathrm{CH_3-CH^+-CH(CH_3)_2\longrightarrow CH_3CH_2-C^+(CH_3)_2}.
$$
The resulting tertiary <carbocation> is the tert-amyl <electrophile>. An aromatic $\pi$ bond of the dimethoxybenzene attacks that <carbon>, producing a <Wheland intermediate>. Removal of the <proton> from the attacked ring <carbon> restores <aromaticity> and regenerates acid. Repeating this <Friedel-Crafts alkylation> at the symmetry-related second site gives the disubstituted product.
The <hydride> migration explains the new ethyl group and the absence of a benzylic <methine group> <proton>. Simply attaching the original <alcohol> skeleton at its OH-bearing <carbon> would not explain the twelve-proton <NMR singlet>.
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