the mulch doesn't look too pleasant. So, I'm wondering if I can turn the
soil over and till the mulch into the beds and then mulch again on top, or
should I rake up all the mulch that's there now and replace it with new?

Your screen name intrigues me. I received my MSW in 1969. What kind of work are
you doing? You will find gardening very good burnout insurance.
You can either even the mulch out with a rake & mulch over it, or till the
mulch into the soil & apply a fresh layer. As it breaks down, the old mulch
will improve your soil.

Iris,
Central NY, Zone 5a, Sunset Zone 40
"If we see light at the end of the tunnel, It's the light of the oncoming
train."
Robert Lowell (1917-1977)