A couple of tips:
Use the Journey Planner on tfl.gov.uk. Its 'average' pace is actually slower than my slow pace, so the default search actually builds in a bit of leeway for most people. Unlike some other planners, it will automatically disregard routes that take you along a section of track that is closed etc - so it will give you a route that currently works. (You can plug in dates and it will factor in planned closures for then.)
Download Bus London and Tube Map onto your smartphone. (Assuming you have a smart phone?) Also, WhatsApp if you don't already have it.
The Underground is pretty easy to work out, with a paper map or a map app. The only trick you need is to check planned closures for the weekend you're down. It's too early to check now - TfL publish a 6 month look ahead. Nearer the time, go onto the TfL website and input your dates: tfl.gov.uk/tube-dlr-overground/status/
Buses are harder. I would avoid buses unless you have no other choice, given your anxiety. When you have to get a bus, use the app to check what time your next bus is coming. If there are no buses on the app, that bus stop has been diverted (this is why having an app is a jolly good good idea).
Download Uber. There's no substitute for a black cab driver's Knowledge, but if you're just doing an uncomplicated A to B journey, an Uber driver will plug in your postcodes and happily zoom off. The Knowledge is for when you're stuck in traffic in central London and need a driver who knows if he takes a left down that side street and a right, there's a quiet street he can drive down and bypass that big jam, getting you to your destination on tim.
Uber is usually cheaper than a black cab, but if there's anything major on like a tube strike, the 'surge' pricing is ridiculous. You'll know before you book an Uber though.
Do not drive. If you're this anxious trying to plan a bus journey, you are not ready to drive on London roads!
Make a plan for what happens if you get separated. You say your DP is even more anxious than you, so if you are both boarding a train and one of you doesn't make it and gets stuck on the platform, then the plan perhaps should be DP stays put (if she is the one who couldn't get on the train) or DP gets off at the next stop and waits (if she is the one who got on the train without you). Agree in advance which person is the person who comes to find the other.
Consider getting tube wifi (it's free for EE, T-Mobile, Orange, Giffgaff, O2, Three, Vodafone and Virgin Media): tfl.gov.uk/campaign/station-wifi
Spend a few minutes setting up tube wifi together when you first arrive in London, that way if you do get separated, you can send an iMessage (if you both have Apple phones) or WhatsApp (any smartphones will do) and figure out a plan if you get separated. Tube wifi works in stations only, not whilst travelling, but it could feed into your backup plan for what to do if separated.
Enjoy yourself. London is great!