If the plastic tip is hard, it can tear fins. Pet stores sometimes have things that are labelled inappropriately (like the 1/2 gallon betta bowls).
Clean water is definitely more important when you want them to heal a torn fin. You don't necessarily need to do more water changes if you have been following a proper schedule up to now.
As well, if you haven't, try giving your fish frozen bloodworms for one feeding during the week. The extra protein is good for them, especially when healing fins.
Good luck! :)