Deep sleep draws 0.9mA

Ok, great - that's a bit better I guess. So are you using an ESP01 with the shim?

So that's using A2 and A3 for serial. You could try A2.mode("input");A3.mode("input"); and see if that helps at all?

A0, A10 and A1 are also connected and you could try setting those to inputs.

Unfortunately I managed to short my cheap multimeter, so now it's not saying anything at all...

That can always be a pain if yoy leave your meter in current measurement mode - I've lost track of the times I've done it :) but if you open it up there's usually a fuse inside that you can replace (because it is so common to do).