Can you help write an email requesting a reference from a previous instructor?

I’m currently sending out job applications and some of them want references from past supervisors. 3 of my most recent supervisors won’t be able to give me references due to company policy (all they’re allowed to say is whether I worked for the company and when, which isn’t much help). My next most recent supervisors are 2 of my university professors for whom I worked as a teaching assistant during my junior and senior years. They would be able to give me really good reviews, but I haven’t been in touch with them since I graduated in 2010. I’m sure they still remember me, but I don’t want to just send them a random message out of the blue saying “hey, can you give me a reference?” I’ve come up with the following message to send them but somehow it just doesn’t have the “right ring” to it so I was hoping the collective would be able to give me some pointers and help me tweak the message a bit. I don’t want it to sound forceful, like I’m not giving them the option to decline. Here’s what I have so far:

Hi (instructor’s name),

How are you? How is everything at SU? I hope this message finds you well.

I don’t know if you still remember me, but I graduated with the Hospitality Management class of 2010 and was also a peer advisor and teaching assistant in HPM (our department).
The reason I am contacting you is because I am applying for a (____) position with (company name) and I would like to know if I could include you as one of my references. This is for an online application and it asks for the reference’s first and last names, title, employer, the nature of our relationship, how long we’ve known each other, phone number, email address and current location, all of which I will fill out once I get the okay from you and they will then contact you via phone or email with further instructions. If this is something you wouldn’t mind doing, please let me know if you would like me to send you anything (i.e. my resume or any other supporting documents) that would help you with give me an accurate recommendation.

Thank you so much, I look forward to hearing from you.

Warm regards,
Jordy

Thank you all in advance, I look forward to seeing what you guys help me come up with!