I'm Miles from the Social Care team at Shopify. Congratulations on launching your store! It's a truly amazing experience getting your first real sales. I'm sorry to hear the excitement was lowered because of the shipping issues.
Without a link to your store, I'll do my best to help you out. If my suggestions don't work, please feel free to reply to this thread with a link to your storefront so I can test the checkout for you.
If some of your customers are receiving free shipping, you might have a free shipping rate set up. If you head over to Settings > Shipping and Delivery > Manage rates, you'll be able to see your current rates that you've set up.
Free shipping can be a great reward for your customers for spending more in your store. You can look at a tiered approach like free shipping if they spend over $100 and then set a flat rate for anything under $100. This can also work for weight based rates (where shipping is charged based on how heavy the items in your cart are).
If you're setting up your own rates, depending on what you're selling, it might be best to set up price based rates so you can reward customers for spending more, as discussed above. If this isn't something you're interested in, then weight based rates might be a better option for you, so the pricing is set based on how heavy the package will be.
Another option for you is Shopify Shipping. If you're located in the US, Canada or Australia, you'll be able to set up live shipping rates for your customers. This mean, you'll be charged the same rate your customers see at the checkout, based on the courier you would like to use. You can also choose to add a fixed amount or handling fee to the prices displayed to your customers too.
If you're setting up Shopify Shipping, make sure the weights of your products are accurate to avoid any extra charges from your courier. As mentioned above, you'll only ever pay what your customers pay for shipping, unless you've added a handling fee or charge a bit extra on the displayed pricing.
Thanks for linking your store. I've just done some test checkouts and was not able to see any offers for free shipping. I'm hoping that means you've managed to resolve the issues.
If you're still seeing the issues, if you could send through a screenshot of your shipping settings, that would be super helpful. Otherwise, be sure to check out our guide on setting up shipping rates as I suggested above.
I also just wanted to say, I love your site! You can tell you've put in a lot of hard work. There's a couple of things that I would suggest taking a look at to take your page to the next level.
Your footer (the bottom of your page) doesn't have much information in it. Have a think about adding a menu there, or even some store policies to help build trust with your customers. Check out our guide on setting up store policies to help you get started.
Have a think about adding an About Us page. This is a great way to build an emotional connection with your customers so they can learn about your story. Think of things like; why did you decide to start this business, who are you and what should your customers expect?
I'd recommend checking out Shopify Compass as well. Shopify Compass provides you with free webinars and courses on things like store design, marketing and more.