You can buy 120 film that has had the edges of the spools machined (or simply cut) off, but the spindle hole in the end is a different size on 120 and the 620 cameras sometimes will not advance the 120 film if one of those spools is used as a take-up, some. I suggest re-spooling as well - it's pretty straightforward.
I just gave away at least a dozen 620 cameras from the 40's through the 60's, including a whole bunch of Brownie Hawkeyes. They will take the 120 rolls as long as you have a 620 take-up spool. Some cameras (most, I've found) will not fit a full size 120 spool to turn in the space provided. Try fitting a roll of 120 into the camera without actually loading it. If it fits and turns without binding, all you need is a 620 take-up reel. Just remember to ask for the spool back if you have the film processed by someone else!