So a few things -
First, there isn't anal herpes and genital herpes. It's all one region, clinically speaking, in regards to herpes. Herpes infects nerve groups, and genital herpes infects the sacral ganglia, meaning it affects the area that boxer shorts cover.
This means that anal herpes is genital herpes and it's all the same infection. You happen to get outbreaks on your anus, but you could get them anywhere in your anal and genital area.
It also means you shed the virus from this area. The virus sheds from the mucus membranes - the thinner, moister skin. For women, this is the labia, vagina, anus and perineum.
So the day after your outbreak, you could still be infectious, as the virus sheds more around outbreaks.
Does your new partner know you have this? Have they ever tested for herpes? Ever had a cold sore? If they have hsv1 already, even orally, the chances that they'd get it genitally are very slim.
Even if they haven't had a cold sore they can remember, they might still have it - 90% of those that have it orally never get a cold sore.
Even if they don't have it, chances are good that you'll never transmit it. Ghsv1 rarely sheds and rarely recurs, so transmission is very low. We don't have transmission studies for ghsv1 like we do ghsv2, but we do know the shedding rates:
Shedding rates: (and you can find all this in the herpes handbook - https://westoverheights.com/herpes/the-updated-herpes-handbook/)
HSV 2 genital 15-30% of days evaluated
HSV 1 genital 3-5% of days evaluated
HSV 1 oral 25% of days evaluated
HSV 2 oral 1% of days evaluated
So for transmission rates. We don't know the exact transmission rates for ghsv1, but here are the rates for ghsv2, and if your math is better than mine, maybe you can extrapolate some figures.
Ghsv2 transmission, female to male, over the course of a year, assuming sex 2-3 times a week:
Only avoiding sex during an outbreak - 4-5%
Adding condoms OR daily suppression - 2-3%
Adding condoms AND daily suppression - 1-2%
Have you and your new partner discussed STDs? Do you know if there is something they have that you could get?