As the title says...

I have the Cokin screw mount with the large disc attached, and attached to that is the large square filter holder.

I just find it a bit awkward to use.

Is there an alternative that might be easier?

I have a set of Lee filters, basically the same as Corkin i.e. filter holder and adapter ring but they are bigger at 100 mm not of 84mm. I like how the filter holder fits onto the adapter ring. There is a grove around the outside circumference of the adapter ring and the filter holder has two plastic lugs which fit into the grove there is then a spring loaded brass lug which can be retracted by a pin. You simply put the holder over the adapter making sure the plastic lugs are in place then let go of the pin and brass lug slips into the grove holing the filter in place.

The filter slots are screwed onto the holder so you can add or remove slots if you require to fit more filters. At the moment I have a set of hard ND grads, a 2 stop soft ND grad, a set of mist filters and a big stopper.

They are a pain to use at first but if you persist then it becomes much easier and I probably never take a landscape now without using some form of filter.