My suggestions are to get a diagnosis, an OT evaluation, and push for a 504 plan. If your insurance will pay for it (or if you are independently wealthy), OT is a big help.
My ds also has difficulty writing. He is also finishing 2nd grade. In K and 1st, his handwriting was poor but legible. He tended to write simplistically, but that's was okay for those grades. In 2nd, they went to paper with smaller lines or no lines, and his handwriting became worse. Also, it began to affect his writing grade (not failing because our sd doesn't give grades for the lower grades, but "having difficulty meeting grade-level expectations," the equivalent of a D). His pediatrician diagnosed him as having a "graphomotor impairment."
Now we have 2 key pieces in place, a diagnosis and bad grades. Those two together allow ds to qualify for a 504 plan, which gives him accomodations, including lined paper. And it's not just a suggestion to his teacher: legally, they have to give him lined paper.
My ds does not qualify for an IEP (under IDEA) because there is not a discrepancy in his IQ and his performance on achievement tests. There are subtests on the achievement tests that might indicate a problem, but other subtests bring up his scores.
My plan this summer is to work on cursive, continue see the Occupational Therapist (okay, we're not rich, but if we pay in advance, we pay $85 an hour (rather than $160), and my younger child starts K this year which reduces our daycare costs), and work on his typing speed. One way I get him to write is to write letters to places asking for information (a map of Sea World, a brochure about Kansas, info on a state park, etc). That way he gets stuff for his efforts.