Most local engineers will start their fees around $750 to $1,000. You might also need an architectural designer (drafter) to make field measurements prepare a site plan for permitting. Some designer will have their own engineers that they work with

I'm an LA-based structural engineer. Since you're in SoCal, the patio cover also needs to meet the building code seismic safety requirements. You don't want it coming down during an earthquake. This is something that the building department will need review, approve, and inspect. You should hire an engineer for this type of work

You should hire an engineer to help you with the beam/post design and with the plans for permits. In general, a 4x10 beam would be more efficient than a 6x6 beam, but the actual size will depend on the roofing material, roof slope, and whether it needs to be designed for snow load. Using a higher grade/species lumber would also be an option to reduce the beam size.

With a 4x beam, 4x4 posts should be adequate, unless they're pretty tall. If you go with a 6x beam, 4x6 or 6x6 posts would be used, to match the beam width.

The existing foundation might be undersized to support increased post loads, and this would be another thing that your engineer would help you verify