I'd be less concerned about if there are spores in the filter media and look more closely at what is causing your algae. You can do all the work to get rid of what you have but unless you change the conditions that brought it about you'll probably get the same algae problem in your new tank.
The kinds of things you might need to look at are your lighting (intensity, color temperature and exposure to sunlight) and possibly adding plants and algae eaters to the tank to outcompete the algae for nutrients and keep it under control. The light received by your substrate can also be a factor. If you have too much of it, it can contribute to providing an abundant food source for the algae and can get too much light due to it going higher than the trim at the bottom of the tank - the combination being a potential recipe for out of control algae on the substrate.
Different types of algae can have different solutions so it would be good if you could provide more information about what you have so others can help you resolve the problem.
By all means set up the new tank (the bigger size has other benefits
) but work on resolving the underlying issues.
Read some of the other threads about algae control. Tim (prov356) provides very sage advice about providing long term solutions to this problem.