1997 Tahoe - Starter, or solenoid, or both?

[72Blazerod]

I have a question about my 1997 Tahoe 5.7 liter 4x4. It seems to not want to start (especially when it is cold). When I turn the key it makes a single click which sounds like the starter solenoid is engaging but then nothing. No cranking. After I try this several times (sometimes 2 times, sometimes 8-10), it will begin to crank over very slowly, then catch and start. Has anybody experienced a similar problem? What do you think it might be - the starter or the solenoid or both? BTW, I have replaced the battery in May with an Optima red top, and I replaced the alternator about 9-10 months ago. This problem began intermittently just before the summer, and I had the ignition system checked and they gave me a green light for the battery, alternator, and starter. But not the problem is getting worse and I want to fix it. What do you think?

Thanks for your help,

Rod