Possibly you've accidentally damaged it somehow with the reset/download process. Or has it fallen into any kind of liquid, intense heat or high humidity lately?
Icons perpetually on/off CHECK
Pixels malfunctioning CHECK
I guess that's the problem.
I've heard reports of Tamagotchi which constantly have a certain pixel either not working or constantly on, perhaps your Tamagotchi has many malfunctioning pixels and the character moves, creating the squiggly lines.
Try changing the battery, as GotchiGirl96 said.
As a last resort, take the whole thing apart and put it back together again - this has helped mine before but as it could result in further damage should only be used for a totally broken Tamagotchi.