Well I'm going to go with transmission problems (yeah great help right)
if you had simply said that it takes a while to engage then I would say that you are having valve body problems or problems feeding the low reverse clutch. Given that it impacts the engine in a way that almost sounds like it is applying too much load (possibly binding one way clutch) to the engine at idle (not sure why the throttle would be unresponsive) because something is binding, once it spins enough to get it in the right place then the transmission is in neutral waiting for things to completely engage and pressure to build (there is a delay from the factory that gets more notable when components wear) and then it works fine.
I'm no transmission tech but that is my semi educated guess of the problem.
Not that knowing that helps you much but there are a couple things that you can do to confirm the diagnosis, all of which will result in you getting transmission fluid all over you and whatever you park over.
First pull the transmission pan, look at the level of black funk & sludge on the bottom of the pan. Some is normal like a thin layer just enough to make the pan look grey with a bit more around the magnets. if the magnets look like you stuck them in a box of iron filings then prepare for a new transmission.
Second is to pull the valve body off and clean it completely yeah you just had it flushed and if they power flushed it it is likely that there is junk helping bind a piston in the valve body thus sending fluid where it's not wanted until the valve unbinds each time. The valve body is all metal so once you get it out of the truck you can use brake parts cleaner to clean it up along with a small brush. I will suggest carefully removing each of the piston sets, one at a time and cleaning the bore and pistons. Pay careful attention to how they came out so you can put them back in the same order. (if not you'll have bigger problems)
If neither of these fix the problem then you're toast.
Oh and one last bit double check and make sure that the filter is seated in the pump correctly and the seal is good.
