Like at cbhours said you dont need much to be able to multi stream. (There's a bunch of ways to do it that dont consume much in terms of system resources, one is the obs rmtp splitter plugin that was linked, another would be the shinebrowser if your sites are supported (bonga,chaturbate,mfc,stripchat,camsoda,cam4 are the supported sites atm). From what I remember using obs on SM gives you a worse placement than using their web encoder (not sure if thats still the case though). Personally I like the shine browser because it integrates all the chats into a single window and switches other sites to spy mode or offline when you accept a private show.
The main limitation for multi streaming is going to be your internet confection, using shine browser can allow you to stream to more sites with your contention since you only push 1 stream to their server and they split the stream on their end. But an internet upgrade wouldn't hurt since it gives you more options (like if there's an issue with shine you can split the stream yourself and still work all your sites) and if you get into content creation faster uploads is always a good thing.
If you are making content or thinking about it, that's going to be a lot more recourse intensive than streaming so you would be looking for a laptop that can do that as the primary focus. For video editing a minimum of a ryzen 5 (a relatively new one, so like 5th or 6th series 7 if that's out) or Intel i5 (just make sure it's a relativly new i5, sites like amazon have people selling i5's from like 10 years ago at the price of a new laptop because people just see the i5 and assume an i5 is an i5), there's also low power variants that run slower, so you would be looking for a sku that ends with h or hq (I think thats the high power sku but I might be mistaken), anything that ends with "u" is a low power variant. you can check things like clock speeds and the release date of the processor on ark.intel.com (just copy and paste in the cpu name which would be something like "i5 12600h" and it brings up the data sheet. for ram you would want as much as possible, minimum 8gb, ideally 16-32gb, and a dedicated gpu if your video editing software supports hardware encoding. These are very general though, best thing to do is look at the recommended spec for the most demanding piece of software you use and then set that as a base for what you are looking for, a little higher if you want to keep it running smooth over the next couple years of major updates. Again that's for content creation, I'm kind of assuming that if you are camming then you probably want to get into videos at some point so better to get a system that can do that now rather than having to buy a whole new laptop.
A question though, what is the main reason you are looking to upgrade? knowing where your setup is under preforming can help you decide on what to prioritize when doing an upgrade or it could be that the issue isnt with your set up and an upgrade wouldn't bring the benefit you were expecting.
