No start

Hello I have a 1996 Chrysler lhs 3.5 with 180,000 klm. My problem started a week ago. When I went to start the car it was slow to roll over and barely started but did. Got it home and checked the battery voltage and it was good but put the battery on charge anyway. Cleaned the battery posts and cables and made sure the ground cable was good.
When I went to start it nothing. Everything worked on the on position but when you put it to start the lights on the radio and heater would go out and the starter would not engage.
Removed the starter and cleaned all the contacts and also bench tested the starter. By putting 12 volts to the starter I got it to engage. Put it back in but still the same problem. Tested the started relay(it does click when trying to start the car) tested OK but changed it anyway. Same results.
I can make the starter engage if I pull the green push on wire for the solenoid and give it 12 volts direct.
I am at the end of what to do, any help would be great.
Thank you in advance Ivan.