I would personally be extremely sceptical of any supplier telling me they were offering FREE SHIPPING. It's just a fact, it costs money to ship items. Can you imagine a supplier asking the Post Office if they could offer Free Shipping?
That's not going to happen, so why expect a supplier to cover the cost they have to pay for shipping your order to your customer?
As they say, there is no such thing as a free lunch and whenever you do see some Free Shipping offer being made by a supplier, you can be sure the cost of shipping that item has been hidden in the price somewhere to cover their loss.
In regards to small profit margins on Drop-Shipped products, that's extremely common. While Drop-Shipping offers certain advantages, good profit margins usually aren't one of them. That's more an advantage of Wholesale, being rewarded for spending more money upfront and taking the risk.
Drop-Shipping is about eliminating the risk and therefore, the rewards in regards to profit margin aren't there. There are also multiple other factors that impact on Drop-shipping margins, but I have gone into those many times before, so I won't expand on those now.
The biggest profit margins you will usually find on Drop-shipped items are from manufacturers who offer Drop-Shipping accounts. They can offer larger margins on their products because there is no middlemen operation taking their slice before it gets to you.
The more links in the chain you can eliminate between you and the actual product manufacturer, the more chance of getting a much better profit margin on the product.
Of course, not every manufacturer offers Drop-shipping directly to the general public, so it's not always possible to get too close to the source. What you will find is that manufacturers who do offer Drop-Shipping accounts are usually smaller boutique suppliers.
Offering Drop-Ship accounts to the general public can be a great opportunity for boutique product suppliers. In essence, it is a way for them to hire their own sales team on 100% commission. Let's face it, that is essentially what Drop-shipping is all about.
The benefit for the manufacturer is that they get a sales team working for them on commission, so no financial risk and no cash outlay. The benefit for the actually Drop-shipper themselves is that they are given the opportunity to make money.
So in the right circumstance, it can be a very beneficial outcome for both parties.