I looked into this myself a couple years back when my own situation changed, and the honest answer is it depends entirely on your timeline. You need to find out what rate you'd actually qualify for today - not the advertised ones, the real offer from a lender - and then calculate how much your monthly payment would drop. Once you know that number, divide your closing costs by the monthly savings and you'll get your break-even in months. If you're planning to stay in the house well past that point, it's worth doing. If you're thinking you might move or refinance again in the next few years, probably not.
The trickier part is that "rates dropping a bit" could mean anything from a quarter percent to a full percentage point, and that makes a huge difference in whether it pencils out. At 6.2% you're not in terrible shape compared to what some folks locked in, but if lenders are offering something like 5.5% or lower right now, the monthly savings could be substantial enough to make closing costs fade into the background pretty quick. You've gotta actually talk to a couple lenders and get real quotes - not online ads, but actual pre-qualification numbers - to know if this makes sense for you. The paperwork is annoying but it's not the limiting factor here. The limiting factor is whether the math works for your specific situation and how long you plan to stay put.