What's wrong with my transmission

I bought a 2004 Chevy trailblazer ls 4.2l 4×4 3
months ago. It's an ex police car and had
289,000 but now has 293,000  When I put it in
drive it takes for ever to pick up speed and
sounds like a (manual when releasing the
clutch in 2nd or 3rd gear and giving gas to take
off). If I manually shift, it won't catch 1st gear
it only runs in (2nd and 3rd) and when I shift to
D it won't catch 4th. Everything else works fine.
If I leave it idling it will over heat to 260 after a
while but once I take off it drops down to 230/
240 an stays there as long as I'm driving. When
I power on my ac it over heats way faster. I
can't afford a new transmission at the
moment. I'm a diy dad an don't trust mechanic
shops for S**T I've been screwed over an
cheated way to many times. So all I'm asking
is for advice. I can't find anything on YouTube
or Google to help solve my problems. TYIA god
bless