The turn signal lights on my 2004 Ford Explorer stay on and don't blink. My hazards flash, but there is no clicking sound of normal use. Eventually, they tend to come back on, or blink with the clicking sound then stay on until I turn them off.

Sounds like your flasher relay is broken. A lot of people think that because their hazard flashers work that everything is OK, this isn't so. Hazard flashers are on a seperate system, and as such they have their own relay.
You will need to find where your indicator relay is located and replace it with the correct type.