How should a new professor sign their name in emails to students?
Sign as you wish to be addressed by the recipient. When I sign an email with my first name, I am giving the recipient permission to address me by my first name. I almost always sign off with my first name, but in the rare instances that I don't want them to address me that way and do want to retain more formality, I use "Dr Firstname Lastname" instead. If I'm really feeling ambivalent, I'll use "Firstname Lastname", which I feel leaves the decision up to them.

I usually also have my full name with title and contact details in the sig at the bottom of my email, separated from my sign-off by a line containing only "--". This is so my contact details are handy, but also for clarity -- even if I am giving them permission to use my first name, it is sometimes useful for them to know my position and qualifications.