You don't need something like shopify unless you intend to sell goods online.
If you like your current website and only want to be able to make small changes yourself, you may want to ask your current web developer about being able to do that. I don't know the specifics of your website but there may already be a pre-programmed back end that your web developer is using to do just that. If there's not, there's going to be no simple way to make small edits (maybe text but that would require basic HTML knowledge and the hope that the website has good CSS). Typically speaking, the ability for the customer to make edits themselves needs to be stated from the onset so the services and website can be tailored to meet that expectation. It's about as much work to add that functionality as it is to create a new website, which is why I hope your website already has it.
Moving to a DIY service can be very time consuming. Typically you won't be able to carry over your current website as it's not designed around the platform's specific template language. Shopify for example uses liquid code to enable the DIY functionality you request. You could pay someone to take your existing website and make it work with one of these platforms but again you are talking $$$.
You could start from scratch and use one of their provided templates but even then it's a lot of time. Those services make their money off apps, customization of the standard template (you'll often find basic features missing), and sub fees. If you aren't happy with the current design of your website I'd recommended just hiring someone to make it to your liking and ensure that it has a backend that enables you to make easy edits.