Grief is a really complex thing and affects everyone in different ways, and I can see how both your dad is reacting and how you are reacting has been directed and exacerbated by grief and circumstance.
There isn't a right, easy, or one-size-fits-all way of coping with loss, especially sudden loss of a such an important person in your lives, and I can imagine that your dads choice to get into dating and moving forward so quickly is one of his ways of trying to cope with the grief, by ignoring part of it and trying to distance his current life and situation as far away as he can from the situation he was in previously, it probably helps that this different and new relationship gives him something to focus on, rather than if it wasn't there, and the stark reminder of what he did have that now isn't physically there anymore (I say physically, because of course the memory and the presence never truly leaves us). However, that is obviously a different way to how you want and need to process this grief, which is totally understandable because everyone is different, and your relationship was different too. Could you have begun, however consiciously or unconsiciously, associating the feelings you have about how abrasive you found your dads new dating and relationship, with some of your own feelings of grief and loss, and it's starting to transfer onto your thoughts about this woman (to the point you are physically reacting and freezing up), and your relationship with your dad. If you haven't already, maybe it would be time to consider some counselling, so you could start to focus more on your feelings and grief with someone completely seperate from the situation, and without any added pressure from sibilings etc about meeting this woman?
I'd also say, that whilst it is important you take as much time as you need, you know you, but it's been 10 years since my mum passed away, and it doesn't really get easier, it just changes, and things will bring it back to the surface, so meeting this new woman now, in 6 months, or in 6 years, might not be any easier, you might just be postponing the enevitable, and that's a choice you'll have to make, but it still is your choice.
Sorry for your loss
, it's shit that you're now in this sitation whilst everything is still so raw.