It’s still good to protect yourself as much as possible (skin cancer isn’t limited to your face)! I would consider UPF/sun-protective clothing if you don’t like wearing sunscreen on your whole body (I sure don’t!). I just ordered some of these protective arm sleeves, and am planning on investing in some leggings as well.
It does partially depend on what time you take your walks/where you live though, as well as how fair you are. If the UV index is below 2 when you’re out, you shouldn’t really have to worry too much. If it’s higher and/or you’re very fair, definitely make sure to protect yourself!
Another option to consider (nylon): https://www.amazon.com/dp/B07X5X541Q/ref=cm_sw_r_cp_apa_glt_i_X524FSM795XRX26PWVWV?_encoding=UTF8&psc=1