If your son is only 5 years old, he still has plenty of time to work on his writing abilities. He is quite young, and in the early stages... If you are really concerned (which it sounds like you are), you should request that the school district assess all areas of suspected disability (if they haven't already). They can provide him with the extra support and tools he needs to gain confidence in his legibility.
I would discourage him using a keyboard at this time (outside of playing educational games on the computer etc.) and rather encourage him
To write and praise his efforts. The idea is getting him motivated to practice and work on his skills.
As to him speaking his responses rather than sign them, that seems to be fairly common among children who have oral abilities. It's great that you're still signing with him though, because as you mentioned there is a huge difference between expressive and receptive language for a deaf child. You could try and encourage him to sign with you (so you can practice your skills!), and maybe set up times during the day where you guys only communicate through sign.
I would definitely pursue and address your concerns with the local school district though. What grade is he in?