hesitating going into reverse

My daughter's 96 3.8l mustang has a hard time going into reverse. This is an automatic with overdrive. The shift lever moves freely, but when you put it in reverse it just stays like it is still in park. It used to take maybe 10 seconds for it to catch,but now i have to give it a little gas while keeping my foot on the brake for it to engage. When i park it on a hill with the front facing up it seems to work almost perfect. It's only on level ground when this happens. Fluid is full, and just recently changed. Other than reverse it runs just fine in drive. I need lots of help. Auto transmissions are very new to me. I've always and still only have manual cars

everything was replaced, flushed, cleaned, etc. even put some Lucas trans fix. like I said everything else works perfect. Put it in drive from parked...goes right in. Thanks for the suggestion

upgate

guess I should update this since there is still some interest. should have
done it anyways. sorry guys. anyways ended up having to replace the tranny.
don't know what the exact problem was, but a very trusted mechanic said
it was basically cheaper and more guaranteed to be fixed to just
replace it. it did have 180,000 miles on it