  We study the processes $e^+e^-\to\omega\chi_{bJ}(1P)$ ($J$ = 0, 1, or 2)
using samples at center-of-mass energies $\sqrt{s}$ = 10.701, 10.745, and
10.805 GeV, corresponding to 1.6, 9.8, and 4.7 fb$^{-1}$ of integrated
luminosity, respectively. These data were collected with the Belle II detector
during special operations of the SuperKEKB collider above the $\Upsilon(4S)$
resonance. We report the first observation of $\omega\chi_{bJ}(1P)$ signals at
$\sqrt{s}$ = 10.745 GeV. By combining Belle II data with Belle results at
$\sqrt{s}$ = 10.867 GeV, we find energy dependencies of the Born cross sections
for $e^+e^-\to \omega\chi_{b1,b2}(1P)$ to be consistent with the shape of the
$\Upsilon(10753)$ state. These data indicate that the internal structures of
the $\Upsilon(10753)$ and $\Upsilon(10860)$ states may differ. Including data
at $\sqrt{s}$ = 10.653 GeV, we also search for the bottomonium equivalent of
the $X(3872)$ state decaying into $\omega\Upsilon(1S)$. No significant signal
is observed for masses between 10.45 and 10.65 GeV/$c^2$.
