Spelling, grammar and handwriting are important. The best thing you can do is to work on it. Buy a dictionary, (print or electronic) and use it a lot. The more you use it the better you will get. Work on your handwriting. Not everything can be typed. Slow down and develop a handwriting or printing that other people can read. Take an English course and relearn the spelling and grammar you should have learned in school. If you were anything like me you didn't pay attention in that class and got the marks to prove it.
As others have said poor handwriting and grammatical errors just scream illiterate, uneducated and incompetent. You will never be perfect at it but keeping errors to a minimum will help you get and keep a job, defend yourself if need be and be more professional.
It is good that you recognize the problem and are all ready taking steps to remedy your situation.
Good luck.