How can I change who the email sender is?
I think you could also do it by a simple filter or plugin: https://enriquechavez.co/change-email-sender-wordpress/
No need to use SMTP.
However these filters will change sender/e-mail for all e-mails, not only Anspress, which could have impact on other plugins, i.e. WooCommerce or Gravity Forms, which is not so great.
I would use it with caution.
It doesn’t come with the option to change the from addy like DWQA. Have send from plugins worked with this?
Cagliostro commented on answer
Just use this for all your need, change the From, debug logs, and more.
https://wordpress.org/plugins/postman-smtp/