Can I offer First Class Shipping with my store's shipping service?

[KellyTour]

Hello,

I am investigating using Shopify Shipping for my store.  I am aware that I cannot use the service for USPS Flat Rate Shipping, instead I must enter product weights and box dimensions for Shopify to calculate the rate.  What has me very confused is that I cannot get rates for First Class Shipping.  If I do a test order, Priority Mail is the only option and it is more expensive.  For example I sell jewelry that is 8 oz packaged in the small flat rate USPS box.  The domestic rate is between $4.80 to $5.20.  Same product weight and dimensions mailed via domestic Priority Mail is $7.90.  Obviously the price difference is problematic to close a sale.  Can you let me know if I cannot offer First Class Shipping with the Shopify Shipping service?  And ideally I would like to offer First Class, Priority and Express domestically and internationally--I have checked all of these options in my zones/services available, but even the Express option does not appear on checkout.  

If Shopify Shipping does not offer me the ability to ship First Class, are there any other freight companies that sync with Shopify that you can recommend?

Thanks, Kelly T