Yes, I probably would do something differently.
First, how did I find out about it? If someone else discovered it and proved it and then told me, but they have not yet published or told anyone else I'd seriously consider immediately killing them (or waiting until they fall asleep and then killing them).
Widespread knowledge that everyone dies every day and is replaced by a perfect copy would be devastating for the world.
Assuming that I'm the only one who knows this (either because I discovered and proved it somehow, or because I just killed everyone else who knows), then I'd probably treat it the same as I would treat finding that I have an incurable illness that is going to painlessly kill me that night. I then have two options:
• Let that happen.
• Die by some other means before then.
In favor of the first option, letting it happen, is that the next day the world has a perfect copy of me that, assuming that most other people do not know that everyone dies and is replaced nightly, the rest of the world believes is me and that behaves identically to how I would have behaved had I survived to that day.
But I have no one that actually depends on me. The world does not actually need me to still be here tomorrow. Some friends would be sad if I were gone, but eventually the line of copies of me is going to end when one of them dies other than by going to sleep, so my friend's copies are eventually going to be saddened by the end of the line of tzs copies anyway.
That brings us to the second option. There are people whose lines of copies are bad for the world. So why not get some guns and take a few of them out? That has a good chance of getting me killed but since I'm dying anyway that night, so what?
There's the moral issue of taking another life, but the people I'd be killing are also going to die anyway that night. I would be preventing a line of copies of them from coming into existence, but preventing something from coming into existence is usually not seen as equivalent to killing something--the widespread acceptance of birth control is evidence of that.
As a practical matter, there would probably not be time enough to actually do this before falling asleep--but my copy tomorrow would know all this too, as would the next copy the day after tomorrow and so on, so could continue with the plan.
It is possible that I (or one of my future copies) would decided that killing is sufficiently wrong that even killing a very bad person who is going to die anyway in a few hours to prevent future bad acts of their future copies is not acceptable.
In that case I'd still consider the second option, but instead of death by going vigilante on evil people I'd probably just start doing things that I've always thought might be fun but had too high a risk of death to consider.
Not quite the same scenario, but the short story "All the Myriad Ways" by Larry Niven covers a kind of similar situation. In that story people develop travel to parallel universes, and discover that the idea that every possible decision creates parallel universes where that decision went different ways is true. The story is about the social implications of that as many people decide that nothing they do actually matters.