Transmission issues... Help!

[GreatDane0512]

Hey Everyone -
I am a new 06 G35x owner and new to the forum.
The car is refusing to shift into 5th gear and occasionally the transmission will overheat and get stuck in 4th gear entirely. I am wondering if this is the valve body /shift solenoid issue I have heard about. I also have no sport mode or manual gear selection. When selecting gears, I can tell the shift cable needs a small adjustment. Not sure if that will fix the problem or not, I kinda doubt it though.
I do know that the car has a junkyard engine but am not sure yet if the transmission is original or not. I also know that the person who installed the engine just barely knew what they were doing. I have already spent some time cleaning up their shoddy work and know that I took a risk buying this car. I'm okay with that but I need some reliability since I am using it as a daily. Has anyone else experienced this before? Is there a common reason why the sport mode may not be working? Thanks to anybody who offers any assistance!