Clarinet doubling on sax?

Currently, I have been playing clarinet for 4 years, and I would describe myself as good, earning a spot in honor band and third chair in class. But lately I was thinking of trying alto sax over the summer, because I’ve heard it isn’t too difficult to transition if you know clarinet well (mostly embouchure is the biggest issue apparently), and I have parents willing to let me rent one and friends who play alto who want to teach me. But I also want to focus on the clarinet as I’ll be doing it in band and marching band likely (and focusing on piano in jazz band since i’ve been playing that for quite a long time ) but at the moment, I don’t know if transitioning would be good for me, and where to start. Thanks for tips.