My thoughts and suggestions
1. You should still be on OPT till 5/31
2. I don't think you are automatically on H-4 since its approval date because you are using your OPT-based EAD.
3. You can file H-1b and at the same time, renew your 485-based EAD in time (usually 4 months before its expiration) as a backup.
4. Consult your company attorney about the issues.