In Thor: Ragnarok, we see Thor and The Valkyrie discuss about his returning to Asgard. She says if Hela is truly there then Asgard is doomed. She also went on to say "Hela draws her power from Asgard as well as you." Is this true and if so what is the case now that Asgard is destroyed?
Hela draws her power from Asgard but not Thor. In the same movie Odin says that the hammer was just a instrument to help you with your powers, the real power in inside of you.
Thor: Ragnarok was perhaps the least comic-accurate Marvel movie to date, however, given that the rest of the MCU has stayed pretty true to the comics, the answer to your question should keep with what we know from the comics.
In the comics, Thor is Asgardian, which means he possesses the above-human strength, speed, endurance, invulnerability, etc. based solely on his physiology. "Thor’s unique physiology of half-Asgardian and Half Elder God grants him the typical powers of an Asgardian God but he is much stronger than the typical members of his race" (Comicvine).
Here's a link to a more in-depth discussion of his powers: https://comicvine.gamespot.com/thor/4005-2268/
Since his abilities are mostly due to this, he should retain all of his natural abilities (including weather manipulation) even after the destruction of Asgard. Mjolnir did assist with some of his natural abilities, and even gave him some new abilities like flight, so with the destruction of Mjolnir, we may see a flightless Thor in Infinity War. But the destruction of Asgard should not have taken away any of Thor's abilities, since they are based on his physiology and not on the existence of Asgard.