Since you're running System 7 there is an easy way to see if an extension or control panel (sometimes referred to as INITs and cdevs by old-timers) is causing a conflict. Start up with the shift key down. You'll want to press it shortly before the Happy Mac appears and keep holding it down until the "Welcome to Macintosh" screen displays. The screen will display a message stating that extensions are off.
If it works fine without extensions, re-introduce each one one at a time until you can pinpoint the cause. When you find the suspect, move it out of the system folder and keep it out, adding the rest back to the system folder (although I'd still recommend the one at a time approach just in case there are two extensions causing problems). Keep in mind that two extensions may also be conflicting with each other, so if you have some extra time it may be worth starting up with different extension sets (which is possible with some of the INIT managers out there, such as Ricardo Batista's Extensions Manager).
If the problem persists without extensions, try booting up from another disk or the Classic's ROM disk (hold down COMMAND-OPTION-X-O while starting up). This will let you know if your system file is corrupt. If it is, find your install disks and re-install the system. If you don't have install disks, Apple provides 7.0.1 and 6.0.8 on their website free of charge. (They also have 7.5.3, but I don't recommend it for a Classic I).
If all else fails, there's a chance it may be a faulty logic board (and if it is, chances are it's capacitor-related). I had a Classic that would intermittently freeze but once I replaced the logic board with a donor from a machine with a bad analog board everything worked fine. (I'm going to re-cap the old one or at least clean it in a dishwasher sometime).
I should also mention that the Moire screen saver is not a "suspect" cdev by any means. It was widely regarded as the best shareware screen saver during its day and I have yet to hear a report of any problems caused by it.
