A sleeping bag liner might help. A cheap cotton one as used for backpacking might help, although the bulk can be a nuisance compared with a silk one.

An alternative is... if you know someone with a thin sleeping bag, borrow it and use at as a bag liner. I use a synthetic bag like that inside another light down bag so I can cope with cool, warm and hot conditions.

If you have any woollen clothing, take it along -- such as a shirt, and socks. Don't wear cotton in those conditions.

I still prefer the groundsheet to be outside the tent floor. If the edges are wider/longer than the tent, then just fold them under so they match the fly and so they don't gather rain.

Look for a mound -- one that stands proud of the surrounding ground. If the rain is stopping, as predicted, these will be obvious as the surrounding ground will be wet.

Use one of the offcut pieces of plastic Bekologist talks about as a doorstep. You can stand on it to remove or put on shoes and not drag dampness into the tent.

The comments on tent ventilation are very sound, but again depends on how much rain the vents will let it if you open them.

Machka and I spent a few very wet days in a Parisian campground in 2007 and even though we had a cheap Quechua tent, it was seamsealed and was totally waterproof, and we followed the principles I've outlined.