Both posts above nailed my view perfectly. I have a Note 4, my wife has an S5. The real difference between the two is screen size and S-Pen. I prefer bigger and love me some S-Pennery. My phone is my life, and it's a serious productivity platform for me, both personally and at work.
She perfers a smaller phone and has no use for the S-Pen features, as she will pull out a laptop for productivity and prefers to do as little as necessary on her phone, so her needs are different.
Both the S5 and the Note 4 have all the stuff Sammy took away recently in the S6/Note 5 lines. Replaceable battery, SD slot, IR blaster, etc. And enough of the reasonably recent features like fingerprint scanner and decent processors to still feel pretty new-ish.
So, depending on your needs/desires for screen size and S-Pen, I'd either do a Note 4 or a S5. Both should be very inexpensive, assuming you can still get your hands on one.
Since you are Verizon, you will pretty much need to buy the Verizon-specific model. If you buy it used, try to find one locally and meet the seller at a local Verizon (not "authorized reseller", actual Verizon corporate) store. Make payment for the phone be in cash and dependent on successful activation of the phone on your Verizon account. While you are in the store, test SMS/MMS, a phone call, and data. Once you know it all works, hand over the cash.
If you can't get it locally, make sure you buy through a reliable reselling channel like swappa. There is a local Facebook group in my area for reselling stuff, and I cringe every time I see a phone on there, because about 50% of the time there are going to be drama-queen posts a few weeks later about how the seller cheated the buyer because it was supposed to be unlocked and either they didn't understand carrier compatibility or the phone really wasn't unlocked (or the IMEI/ESN was registered as stolen)
Don't worry if the battery life isn't too good on a used S5 or Note - because the battery can be replaced very cheaply. I just picked up a pair of Anker replacement batteries for my Note 4 for 30 bucks with a travel charger on Amazon, because both of my batteries are a year old and really showing their age.