If you replaced TL7705 with simple res/cap reset and problem still happens (though its occurrence is statistically reduced) it is a further confirmation that noise you introducing will suffice to reduce signal integrity.
Again bad grounding could be the main reason. A missing pull-up may be the second one, or not connected pin, or again your psu is coming with low voltage or it's quite noised.
Which value of main clock are you using ? Are clock tracks short enough ?
From what I've understood your application is a single chip; no exram, so no ALE has been used. You said an LCD is connected to port. Which port ?
It's very difficult to help without see a full circuit and/or engineerization used. For sure if your LCD is printing funny characters when you touch ground it means int ram contents is destroied or who knows where the program counter is pointing to. Did you make use of watchdog ??