Hi OP, congratulations on passing your test!
I passed in late 2018, in my thirties. The following day I went for a drive to the local halfords to buy some P plates - a couple of miles maybe? Took my Dad with me... and wanted to murder him within the first 2 minutes. We get on brilliantly normally but he does things very differently to how I was taught. I kicked him out of the car at the end of his road on the way back and drove on, on my own, to visit my MIL a mile or so further on. And it was fine! Even though it included the massive roundabout I was terrified of. That night I went out for dinner with a friend - again, only a few miles, but I picked her up and drove us both! Spent the whole time apologising for how bad/slow/nervous I was, but it was fine.
The next day I drove 2 and a half hours from my parents' house (where I'd taken the test) to my own home - all but the first and last couple of miles were on motorways, including the M25. Had my husband in the car with me but he can't drive so to all intents and purposes I was on my own. And again, it was fine!
I'm not a natural driver, at all, I still get nervous about unfamiliar routes, and I worry far too much about what other drivers think of me, and about pissing people off if I make a mistake/hold people up. But the vast majority of people are fine - you'll always get dicks who think they're more busy and important than anyone else on the road, but try to ignore them. 
The best thing you can do is just do it - don't let it build up and up in your head. To start with I would find excuses not to use the car - "oh I'll just walk it, it's good for me" or "the parking might be busy so I can just get the bus" - but the best thing I did was just start driving regularly. Driving to work every day was great for building up my confidence. And it doesn't gave to be long trips - I just popped to the nearby supermarket regularly to pick up things we needed.
Good luck for your driving with your dad this weekend but honestly, I'd try to go out on your own as soon as you can, even if it's just for a mile or two. Try and stop yourself and say "why am I walking to the bus stop? I can get in the car!" You said you can't wait until you can just jump in the car, but you can! Any time you want. You just have that weird "I've passed my test but it's not really OK for me to drive yet" feeling. You can do it, you just need the confidence. And I say this as the person who kept their P plates on for 5 months because I was too nervous to take them off... 