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You might be able to do it with the following workflow:

1. First do the dissolve like you've done, which results in one multi-polygon
2. Now split them out using the 'vector' -> 'geometry tools' -> 'multipart to singleparts' operator
3. Now do a spatial join with 'vector' -> 'data management tools' -> 'join attributes by location' to get your attributes back.

Edit:

For the join by location, you should either use your original points to join to the 'singlepart' features (with a contains operation) or use your buffers (using intersects). You'll only be able to copy attributes from one of the features, where two or more have been merged, of course.