You will be fine. Don't worry.

Did your employer start to apply for new H-1B for you?

Usually, it will take 1-3 months to apply for a new H1. However, most of case will be done within one months. My case was done in 1.5 months. Because our medical school in the biggest one in US and it has many employees.

If you still don't feel comfortible, you can go/call to the Human Resources or International Office every week. Give a little push to them.

Never consider to transfer to H4. Too complicated and waste a lot of time. With H4, you can not work.