Can I Fly Within KSA while waiting for my Iqama to process?

The point of Iqama is to identify you thats it. So same applies for passport. and your answer is yes you can travel within ksa with a passport.

Sorry, I guess I wasn't clear. I will not have my passport while my Iqama is processing, isn't this correct? So I would only have a US driver's license as ID. Will this be sufficient?

I wouldn't advise. You need to have your iqama (local identification card) or your passport. I don't think anyone will ask for identification but why take the risk.

There is no document check for domestic flights so you really don't need them to buy a ticket and to get on the flight and you can fly from Jeddah to Riyadh and back without any ID.
But by law everyone must have his/her ID (iqama or passport) every time with him/her and there are security checkpoints around the towns, on the way to airports and across the country. In big cities where is big traffic the guards are watching to the bypassing travellers and asking for ID very randomly if they see something suspicious, but in remote areas and smaller airports, particularly near the country borders they are checking everyone 100% and very thoroughly.
Therefore you need to have at least a copy of your passport  (including KSA visa page) with an official letter that your passport is submitted to MOFA for iqama issuance process.

take one of them. otherwise you will face problems.

While your passport and iqama is away, for your iqama processing, you will be given a ;letter from your company (in Arabic), mentioning that your iqama is under process.

By law, anyone caught without an iqama for the first time will be fined SAR1,000. Second time SAR3,000 and for the third time, the person will need to spend 3 nights in the prison and possible deportation.

It is best advisable not to travel or take a leeter from your company stating that you would be travelling.

Regards,
Raunak

Hmmm I suppose I'll check with employer once I am there. I am just trying to plan for the first 3 or so months I am there without a passport. I will NEED to get out of my city at some point. And unless I can fly (Riyadh is 9 hour drive), I may be stuck there 

is enough have a letter from your comany mentioning that your Iqama is under processing and a copy of your visa!

Yes. For your 2 main identification documents -passport and iqama are not with you for obvious legal reasons. I have done it before.

Regards,
Raunak

I did fly without iqama and passport.
Though I have a photocopy of my passport with me coming from company  with stamp (Arabic text). <-I suppose this is the letter coming from the company. Lol

By the way if your employer is a big company Iqama issuance doesn't take all that long. You can expect in 2 weeks max...again this is given your sponsor is a big company with you having direct access to concerned HR people.