Hi David, welcome to the forum!
Range is a relative term, and walkie talkies in general are considered "short range’ communications. Considering this, a recommendation for handheld radios is going to depend primarily on the frequencies and service you plan to use. Are you looking for radios to operate on FRS, GMRS, amateur (ham), business band, MURS, or some other service? Wattage will also depend on this, since some services such as FRS and MURS are restricted to 2 watts or less.
Baofeng radios may come pre-programmed to some frequencies that are not legal for transmit in the US and some other countries, such as police, fire and public service bands that you mentioned, so you don’t want to open the box, turn it on and simply choose a pre-programmed channel to use.
You mentioned that you don’t mind a licensed service. Depending on what you plan to use the radios for, it sounds like GMRS is your best option. Most GMRS handhelds are considerably less than 2 watts, but the Wouxun KG-805G is a full 5 watt radio, there are a large number of accessories available for it, it supports repeater channels, which can greatly increase range, and it has a removable antenna, which can increase range even more. A license costs $70 for ten years and covers your entire family, however there is currently a proposal by the FCC to lower it to $50. If approved, that would be even better.
While a pair of KG-805G radios is a little above your $120 range, it offers a lot of features that other GMRS walkie talkies don’t have, which makes it worth the price.