If it stinks and it's had no drainage, it's gone anaerobic, which is bad. If the worms are multiplying it means there's enough air for them to breathe and some useful microbes for them to eat, which is good.

Possibly your best option is to mix that nasty old stuff in with your new bedding (dry) and let it soak up and redistribute all that excess moisture. Make sure the new setup is open enough to allow air flow and water drainage, and do not pack down the bedding. The familiar smell and taste should help to convince the worms their new setup is home (reducing escape attempts), and with the increased oxygenation those smelly anaerobic bacteria should die off. This will leave the good guy aerobic bacteria also in there to take over and expand, providing plentiful food for your worms and no nasty smells for you.

This doesn't just apply to moving from a bin in poor conditions to a new one, trying to clean up bad microbes or rescue sick homeless worms. It's always a good idea to mix some Living Material - that is, something infested with bacteria, fungus and protozoa involved in decomposition - into the bedding and food of a new bin. Worms don't really eat solid food, they suck up the slurry of microbes on food and bedding and anything broken down enough to come with it. You want to get those microbes in there eating and breeding so it's a suitable place for the worms to live and eat.

Hint: don't feed your worms right away. Overfeeding is an easy and common way to kill worms, and to flood the bin (most foodstuffs contain water which is released as it breaks down). As long as it's microbially active and breaking down, they can just eat their bedding to keep going. Give them some time to settle in and rebalance the microbe population before you start giving them richer material to process.

Whatever you happen to use for a worm bin it should never stink. It may have an aroma temporarily just after adding some new food. Aeration is one of the key factors in maintaining a proper worm bin. The balance of brown bedding to green should be substantial. The green will be any food scraps or coffee grounds you might add. If the bin doesnt have enough aeration they wont like and leave,, the same can be said if it is either too wet or too dry. When adding food scraps it would be best if you first chop it up as fine as you can for the decomposition. After all it will be the bacteria involved in that decomp. that the worms will actually eat. The amount of food you add should only be in relation to how many worms are in your bin. If you have for example 1/2 lb of worms then you shoulnt add any more than 1/2 lb of food and dont add any more untill that food is visually gone. Even though there will still be food you cant see you will be able to add more new food in another area of the bin.

I'd start the new bin with plenty of fresh bedding (shreaded cardboard/newspaper) that is moist but not dripping wet. Hummersteve is right, your bin shouldn't stink. I tecommend salvaging your worms and dumping the current smelly mess in your garden.