This is a question about one of my newer cars, a 95 Lebaron Convert. The starter motor is laboring to start the car. The battery sits at 12.78 volts, and then when you crank, the voltage drops to 11.34 within a few cranks. The engine then fires and starts, but it doesn;t sound good in the process. You get the impression that the voltage would have dropped further if the engine had not kicked over. Is this a bad starter, drawing excessively, or is it a weak battery that cannot take the load of the starter? How can I check this out? What test will point me in the right direction. Thanks.

The 11.34 does not seem too bad, perhaps even normal?
Is this voltage at the battery terminals or at the starter?

Many auto parts places will test your battery for free. I have seen lots of good batteries replaced when the problem is actually elsewhere. People often change the battery first because it is the quick and easy thing to do.
Sometimes a new battery will temporarily "cover up" the actual problem, such as a worn and draging starter.

I got a "junk" battery from a place about 8 years ago (someones trade-in battery) when I was looking for a half good battery to allow me to start and run my too-rusty-to-drive Valiant to keep the engine in working condition. (I did not want to buy a new battery just fire up the old car).

Guess what? I sent that car to the junk yard 3 years ago, and today, 8 years after I got that battery out of the junk box it is working fine in my 1979 Dodge truck with 318 V8 that is driven daily.

11.34v while cranking is plenty. From your description I would start with a different starter. If the battery was low, the starter would run slow, not sound bad. Sounds like a sticky solenoid in it.

Do you have a battery out of another car you can try just for experiments sake? Check cabling and terminals while you are at it, often corrosion can creep into a cable without any indication from a quick glance.

All great thoughts. That was voltage at the battery. I will check the voltage at the starter, and see if maybe there is a significant volt drop in the cable. It is an easy starter motor to change, and I don;t mind doing so. It's not much harder than changing the battery if you don;t mind lying on the ground. I would just like to know which to do.