Is your game paused? You can not drag and drop items when the game is paused. Other than that you should be able to drop stuff on the ground or put it in barrels, etc the same as with DD. If your characters are split up, you can have one character open a container, then switch focus to the other character to transfer things into the container from a distance. After the very beginning of the game, though, you should not need to do this.
The easiest way to get rid of excess stuff is to have a summoning doll carry it (the first is available on the level you reach after jumping down the pit in the arena).
Also, with the latest
patch installed (v1.49), or an already updated re-release version of the game (v1.47, probably with the 1.49 patch on the disk as well), the penalty for being encumbered was changed to not being able to move, which is fine for a summoning doll. The game version is shown on the bottom right corner of the main menu.
Summoning dolls can not carry quest items or set/unique items in the act they are found in, but other than that make great mules.
Once you have access to the battlefields in each act, you can visit the BF merchants (repeatedly, if required) to sell excess stuff (summon a doll first to include its inventory in the trade window). The BF merchants can hang onto some of the stuff you sell them for awhile, so some people just sell to the potion dealer, so that the ones who sell jewelry / magical items, weapons and armour do not get their inventories cluttered.
