Salvage An Embarrassing Attempt To Negotiate?

I was going to accept my offer, but my friend told me I was underselling myself. I called up my supervisor who called me with the offer only to dropped the ball. Made a cringe call and shot myself in both the face.

This is how it went:

"So, um, yeah um. Thank you for the offer. So, I was looking around at land o' lakes and its pricier than expect, so um based on this I think the $XXk might be on the lower side, um do you think I can see what the full details of the offer are?"

If I was that hiring manager, I would have pulled the offer right on the spot because of this horrendous, cringe call.

I didn't know how to negotiate because I hadn't received any details outside of the base salary. They were also asking if I was verbally accepting the job but I didn't want to say yes and imply that I agreed to the salary, so I beat around the bush and asked for the full details so I can review them. I don't know if this was also a fuck up or I was supposed to say I accept the offer pending review of the full details and then negotiate after receiving the offer letter. It's possible that they offered a minimum acceptable salary which may have been on the low end, but still not too far below the mid. (Offer is 4k more than what I wrote on the job app as my min)

i placed myself in a corner. i can not ask for anything which will be strange given that i made a big deal about it or i can ask for 2% more. trying to save face is priority

Kind of stupid question. I have a deadline to accept, the offer but I would need to negotiate. Do I send them an e-mail and not sign the digitlal offer or should I sign and then send the e-mail. I can state that I signed as my intent to be employed but not yet in agreeance with the comp package?