Transmission only goes into first and reverse gears

What was the problem with the cooler? If it was overheating the fluid and transmission perhaps there is already do much damage internally to the transmission. I would suggest you drain and remove the pan and see if the filter is clogged and how much debris is inside the pan that fell off clutch material, etc.

We replaced the radiator. We flushed the transmission three times. Changed the filter. Still only goes into reverse and low gear. Could it be the transmission solenoid?

You should retrieve the trouble codes. Since the transmission is electronic, there is most likely a code set.
You never did mention why you replaced the radiator and flushed the transmission. Was it overheating? Did you get second and third gear before doing this work?
It could end up being the Solenoid/Pressure Switch Assembly .
The Solenoid/Pressure Switch Assembly (Fig 1) is external to the transaxle and mounted to the transaxle case. The assembly consists of four solenoids that control hydraulic pressure to the LR/CC, 2/4, OD, and UD friction elements. The reverse clutch is controlled by line pressure from the manual valve in the valve body. The solenoids are contained within the Solenoid/Pressure Switch Assembly, and can only be serviced by replacing the assembly.

Fig 1: Removing Solenoid/Pressure Switch Assembly

The solenoid assembly also contains pressure switches that monitor and send hydraulic circuit information to the PCM/TCM. Likewise, the pressure switches can only be service by replacing the assembly.

The radiator was replaced because the transmission cooler was leaking coolant into the transmission making the transmission fluid looking like Pepto. That is why the transmission was flushed. When that happened we had no gears. After the car ran for about a week. That is when it lost all gears except reverse and low.

Okay, retrieve the trouble codes.