I have an Indian Passport with a US Visa

If those are your only stops and, as you have mentioned, your layover in Shanghai is less than 24 hours long, you are definitely eligible for 24 hour visa free transit and you don't need a visa of any kind. You will just, upon your arrival in Shanghai, approach to 24 hour visa free transit counter where you will apply for a temporary entry permit that will be definitely granted to you to be allowed to collect your luggage and to perform a recheck.

You should print out your tickets and have those printouts in hard copy with you and you will declare at check-in in Mumbai that you will travel to Seattle via Shanghai using 24 hour visa free transit. Basically, this is it. You are definitely fine because separate tickets are accepted for visa free transit within China.

Thanks

So basically, what you are suggesting is that I will be allowed to get my baggage although its a different ticket within 24 hours and they'll let me pass through immigration after getting that temporary transit visa thing. just asn FYI, how much does that cost and what all things would I need?

Yes, you will be allowed to do that. I repeat that a travel on separate tickets is fine and you will be allowed to collect your luggage and perform new check-in after you have applied for a temporary entry permit.

There is no charge for this because this is a temporary entry permit and not a visa, that allows you to perform your necessary formalities in order to successfully accomplish your transit. Of course, in case if your transit in Beijing is a longer one, you are also allowed to make a brief visit to the city. Beside your flight tickets printouts and your valid passport, you really don't need anything.