Accounting/legal service for small part-time consultant

I am starting part-time consulting in the U.S. on the side of my full-time job, mostly for fun (programming and web development). So far, I only completed one project and although I am marketing myself for more, it’s not unlikely I’ll keep hovering around 2 to 3 projects a year.

That was probably a mistake but since this was my first experience, I focused on making the client happy (and making sure I wasn’t violating the terms of my full-time contract) more than on the legal aspects of me effectively taking on a second job.

Now that the job is done and the client is asking me for an invoice, I realize that there is more to the legal side of my fun little consulting gig, and frankly, I’d rather not have to deal with it.

For just a few projects a year I am mostly doing for fun, I don’t want to get deep in the weeds of paperwork, licensing, and the like. And I would not mind if the money I take went down provided someone was dealing with that for me.

Are there any company/services out there that, for some fee (whether a cut of the contract or a flat fee) allow you to be "as carefree as an employee" when it comes to legalities, taxes, invoicing, etc. ?

What I mean is with my "regular" employer, I don’t have to worry about taxes, business licenses, or billing the clients I am working for through my employer.

For my consulting work, is it possible to go through a third party that, for some fee, deals with all those things while leaving me the job of finding clients, negotiating rates and extent of projects, whether I want to take on a project or not, etc.

I realize what I am describing might not be all that clear. Feel free to ask for clarifications if it indeed isn’t.

To clarify, I am not looking for explicit recommendations of a particular professional or company. I am more interested in the right keyword and distinctions between business titles that could carry services (or packages of services) like the ones I am looking for.

Not being from the U.S. originally, I don’t even know where to start. Where I am companies offering similar bundles of services would be called "secrétariat sociaux" or maybe "coopérative d’indépendants" among other names. But I don’t what the "equivalent" is for the U.S.