The main difference between upgrading handset at a big box store vs at the carrier directly would be price. Any new contract terms would set by the carriers, so that will be the same, sparing any sort of sub-contract from a shady third party of online cellular service reseller. Essentially though, places like Radio Shack, Wal Mart, Costco are empowered to offer the same handsets for less money. OTOH, if you look at Verizon prices online at their site, they also run periodic sales on smartphones (etc.). My preference is to deal with carriers directly whenever possible. Since it sounds like you are merely a second line on a shared plan, it should be relatively routine to upgrade, even at Target. Just make certain you are comfortable with whomever your are dealing with before signing up for your upgrade/new contract for your line.

Keep in mind that certain handsets require specific data or messaging plans, that's the only rub, esp. if you are moving from a 'dumb' phone to a smartphone. You can look on the Verizon site for more specifics for each model you are looking at. What's required on your husband's line will be separate from yours based on his having an iPhone. You will share the monthly bucket of minutes with him, regardless of the phone upgrade choice. Data will be a separate 'bucket of bytes' per line.