Hi Dana,
I know you posted this awhile ago but I thought I’d share anyways. I’m an occupational therapist who works with kids in the school districts on writing and penmanship.
The muscles of the hand don’t really finish developing to allow for the precision of writing until 6-7 years old. So with my kids (almost 4 and 2), I’m only doing written work as they are interested with a focus on drawing. When using utensils we almost never use pencils and I always use the fat crayon/markers so their hands don’t have to work so hard.
You should definitely correct errors now and not just let things go. Kids very easily develop bad habits, so why make more work for yourself later. I actually don’t allow my son to use any writing/coloring utensils unless he is using the correct grasp. I literally take the utensil away if he refused to correct it.
Also, you should have your child use an easel to do their written work or coloring. It encourages top to bottom motions and further develops the should, arm, and hand muscles.
The program I like most is called Handwriting Without Tears, they have a great website and I love their materials. I use them almost exclusively at work and will be using them with all my own children.
Hope these help.
Meghan