How non-motile plants always do:
This is a frame challenge, but the answer is still the most viable ACTUAL answer I can think of.
Plants don't have a realistic chance to simply migrate off-planet on their own AND reach someplace likely to support terrestrial plant life without help.
That being said, if anything other than a few bacteria are likely to survive the death of Earth, plants are. Why? Most life needs plants. So the best way to get plants to other habitable planets is to convince humans to take them there.
Plants frequently rely on animal species to carry their seeds on fur, passing through the digestive tract, or just squirreled away.
So the best adaptation a seemingly sentient plant intent could come up with would be to alter human behavior and make people fear dying out on Earth. Humans would colonize planets and build greenhouses. They would build space habitats full of oxygen-generating plants. Heck, humans would make rose gardens just because they're beautiful.
So how to convince humans to do it? Large numbers of plants could die. A few rapid ecological collapses will panic humans. Some kind of mind-altering chemical that makes humans full of existential dread about global climate could work, too. You really only need SOME humans to leave, because they would all carry plants with them.
But Earth will still be a paradise for humans AND plants even if man has to retreat to domed and sealed cities. There's nothing wrong with sewing your wild oats (metaphorically or literally), but I suspect there will always be "No place like home."
- IF NO HUMANS IS ESSENTIAL: then find something valuable enough to get aliens to come to Earth and collect terrestrial plants to move to other planets. I don't know what kind of product aliens in your universe might like, but hopefully your plant conspiracy does. Since you're asking us the motives of adaptive plants, maybe that's it - your plants advertise their presence to aliens via some kind of communication array. Aliens show up so fascinated at receiving a signal from plants that they collect specimens to take home with them.