You're in a bit of a pickle here. The disc on your windscreen doesn't matter much once it's expired - what matters is whether your vehicle tax is actually valid. So if your tax ran out last month and you haven't renewed it yet, you're technically driving an untaxed vehicle, which is an offense. There's no real grace period that I'm aware of, so if you get pulled over, you could absolutely get fined. The DVLA takes this pretty seriously because it's linked to insurance and MOT stuff.
The good news is that you can sort this out pretty quickly online, and once you've done it you can backdate your tax to cover the period you weren't taxed (I think it's usually up to a certain point, but don't quote me on the exact timeframe). The fine for driving without valid tax can be substantial though, so I'd get on it as soon as your insurance renewal comes through. In the meantime, honestly, try to minimise how much you're driving around - I know that's not always practical, but it reduces the chance of getting stopped.
Once you've renewed everything, peel off that old disc and stick the new one on. Just make sure your insurance is actually active before you drive anywhere, because that's a whole other offense if you're caught without that too.