Honda - Accord :: From Reverse To Drive / 2 - 3 Second Pause Before It Engages - Code P0506

I have a 2003 Honda Accord with 207,000 miles on it. It's been doing the following:

- From Reverse to Drive, there’s a 2 - 3 second pause before it engages
- Sometimes when driving at higher speeds (55 - 70mph) then slowing down for a light or an exit, the car will sort of lurch/jerk around 30mph on the slow down. It’s very noticeable.
- Check Engine light has been coming on off for months. I thought it was a gas cap issue but the mechanic said it’s a P0506 code.
- Hard shifting in my day-to-day driving- Just fyi, we had the transmission rebuilt a year and a half ago.

We’ve taken it to the transmission shop twice and they said they don’t feel the things we describe and are getting no codes. We’ve taken it to the Honda dealer twice and the honda mechanic did a software update that didn’t fix the issue. I took it back and then he said he felt the hard shift/delay from R to D and that it is a transmission issue because the engine wouldn’t be engaged at all from R to D. The CE light wasn’t on for that visit and wasn't giving a code but then the CE light came back on when I was on my way back to the transmission shop. I read that there are specific things to do to try to clear the P0506 code (clean the throttle body, check for vacuum leaks, etc.) but our Honda dealer didn’t do any of those things.

Is there an engine issue related to the P0506 code that could make my car do these things but feel like a transmission issue even if it’s not (since I’m not getting any transmission codes)? Could I have two separate issues - whatever is triggering the P0506 code and then whatever is making the car shift hard, slow to engage, and jerk when slowing down? My transmission guy told me since they’re not feeling anything or getting any codes, the only thing left for them to do is take out the transmission. I’m at a loss and feel like I’m wasting their time, but there’s definitely something wrong.