You have the 5R44E transmission and it needs to be rebuilt, it for sure isn't shift solenoids.
Good read here on what people have tried: Help Please - No Reverse or Drive But 1&2 Work - Ford Explorer and Ranger Forums "Serious Explorations"®
But ultimately your money and time would be better spent on a rebuild or swap.
Manual trans vehicles need clutches at $900 a pop if a shop does it and over 10 year period depending on the miles you could need 3 clutches, so $2,700.
Automatics do need to be rebuilt over time and miles, but it does come in one big chunk unlike clutch replacement