Oh, I hope you haven't removed the SWIM yet! If you haven't yet recapped your LC, that's most likely the problem. I repair old Macs as a hobby/side business, and I've never found a LC, LCII, or LCIII that didn't have leaking electrolytic capacitors. They cause every imaginable problem: wobbly/scrambled display, intermittent boots, the "Falling Bomb" sound through the system speaker, random floppy problems, et c. The surface mount electrolytics Apple used in the machines of the time were of inferior quality, and to add to the problem, a system powered on at full load after many years of sitting on a shelf will stress the caps harder, as they're probably in need of reforming.
You should immediately open your LC and check for corrosion or something that looks like partially evaporated cola, especially in the analog area near the back corner, opposite the power supply. Clean any off immediately: in addition to causing electrical problems, the leaked electrolyte also destroys PC board traces, causing them to erode or detach from the board substrate. Wash the area using isopropyl alcohol and Q-tips. You should then replace the electrolytics with new ones: a lot of people recommend tantalum caps, but I like to use aluminum electrolytics so the rebuilt motherboard looks like the original. You can get the replacements from Mouser for less than $10.
When you remove the old caps, don't try to desolder them: there's a good chance the traces' attachment to the board isn't great from the leaked electrolyte. I use sharp, close edge cutters, and cut the caps off near the board. A quick swipe with the soldering iron removed the bit of the cap pin still on the motherboard. Soldering the new caps is usually easy with a traditional iron, just make sure to use a fine tip.
