1992 Olds Ninety-Eight Starting Problems

All,
I bow to your greater knowledge than mine, and ask for suggestions as to cause:
This car has 290K miles on it and I hate to send it to GM heaven over this.
As of late, the car has been hard to crank. When you hit the ignition, for a few scounds, nothing happens. Then, the interior lights dim and the starter kicks in and then the motor begins to turn and after 4-5 pulsating motor cranks the engine starts.
Well, at times, nothing happens when the igntion switch is turned at all. The alternator is operating properly, however, I assumed that the battery may not be accepting a full charge and was really digging for enough amps to crank the car.
I replaced the battery, and the first crank was a little slow, but the engine started fine. After a few minutes, I shut the car off. Now, nothing at all happens when you hit the ignition!
Could this be a bad ignition switch? Once you get the motor to turn over - then you are good to go, but most times all you get when you turn the key is the fuel pump and nothing else.
Any other suggestions? I am in for a CHEAP fix if there is one!
Thanks in advance! BigBob
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over -

I'd go with a starter first.
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-Mike-
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Yup, or bad cable/ground connections.

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Very true. I've had this before. The connection at the starter can corrode and even burn through partially, resulting in a high resistance connection. Shep's advice is good advice.
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