Toyota Tundra Starter Problem

I have a Toyota Tundra 2000 SR5 V8 4.7L with about 100k miles on it. Today the starter did not disengage after I turned the ignition off. I did not disconnect the battery in time and now the battery is completely dead (no charge even to the door locks). About two months ago I replaced a three-year-old battery after I could not start the truck and measured only about 9 volts on the battery with a multimeter. Now I realize that a starter already going bad was probably affecting the first battery.



Questions:



Does it sound right that a starter that does not disengage will kill a battery pretty quickly?



For a starter that does not disengage, are we looking at the starter or the solenoid or the relay or not enough information yet to know?



Does this model Tundra have an integrated starter and solenoid or separate?



Where is the starter?



Is there any recommended order of replacement/troubleshooting (for example, start with the relay, or it’s probably the solenoid so start there).



Thank you for answering my questions.

It appears that prior to 2001 that the solenoid is external. If the starter ran for more than thirty seconds, it might be damaged as a result.
I’d buy the Chilton manual for this truck before I did anything. Get as specific a manual as possible, preferably just for the year your truck was made.
Your ignition switch might also be the source of this problem, not the solenoid or starter. If you have a heavy set of keys that hangs every day from the switch, it is possible that the switch is not cutting power to the starter when you release the ignition. Or even when you shut the key all the way off.
Otherwise, buy the manual and replace the solenoid first and the starter second. The regulator should be internal to the starter.

Thank you very much for you prompt and very helpful reply.