Can we hand out the flowers collected from Survival for Vesnali? Or do we have to stick to the ones we find around the site now?

You can only use the flowers that you find floating. The flowers from Survival are used to leave at the Survivor Memorial.

Survival flowers restock, but they also float, so to be specific each of these two holidays has its own kind of flowers. You will not find Survival Flowers floating now. Sometimes Vesnali flowers retire, in that they don't float anymore. You can still hand those out for Vesnali Coins if you want to (although they tend to go up in value pretty quick so you probably don't want),

If you want to check which ones you can use for Vesnali, you can go to the fansite Subetalodge and check in their Item Daftabase. If you want to collect the new flowers for whatever holiday and to see which ones they are, usually someone makes a thread in the Forums and staff will sticky it.