1-2 cm is what I would call a branch rather than a twig, and not something I would put in a compost because it won't break down in time.
Re the smell and look, it sounds like you don't have a good mix of ingredients in the compost, so one thing you could do is rebuild it. You will need some (a lot of)
carbon material eg dried leaves,
newspaper, autumn residue from the garden (eg spent stalks)... This needs to be mixed or layered with the rotting vege scraps and grass. You need much more dry material to wet. Have a read up on some of the other threads on composting for how to make a good mix. A soggy, smelly compost means there is too much green and wet material in the mix - composts shouldn't be rotting.
The other option would be to make it into a worm farm. You will still need dry material but it might be more manageable this way. What kind of compost structure are you using?
Either way, when you remake it, take out the sticks/branches and do something else with them (hugelkulture). If they are still long this won't be too onerous (or do you have a really big compost?).