Do you have a SmartThings/Aeotec hub, and if so, which one? If you do have one, that opens up some additional options. 
Here are a list of methods for processing webhooks using the 2023 SmartThings architecture.
1) SharpTools
No hub required. Easy visual interface that runs in most browsers. Free trial. Active community that can help you design rules. Costs $30/year for the pro tier.
https://sharptools.io/rule-engine.html
2) Webhook Edge Driver for local LAN communication.
Requires hub. Some technical setup, but not too bad. Doesn’t require anything outside SmartThings,
Although the easiest way to do this is probably with the paid license third party app, SharpTools, if you do have a hub you can also do it with a community-created Edge Driver.
However, there are a couple of restrictions.
Edge Drivers are not allowed to communicate outside of your own network. So if you’re just trying to send a message to another device in your own home, that should work.
[ST Edge] Web Requestor: a driver to issue local POST and GET HTTP requests
3) Use your own server to handle requests to/from the internet
But if you’re trying to connect via the internet, and you don’t want to use any outside services, you need to first communicate to an always-on server device in your own home, then have that handle the webhooks. So this method requires both a hub and a server device. It’s doable, and a community developer has created a project called edgebridge you can use to do that, but obviously it’s much more technically complicated than SharpTools and has its own costs.
4) Add a Hubitat hub and use Webcore there
Hubitat, a competing hub brand to smartthings, has now added Webcore as part of its primary features, and there are some community members here who have created code that will let you communicate between a smartthings account and a hubitat hub, typically creating proxy virtual devices, so that means you can run webcore on hubitat, and have it act on devices connected to smartthings.
Hubitat runs almost everything locally, and does use groovy, but it usually requires more technical expertise to set up than smartthings. And while it has an app, it’s not as polished: some people use Sharptools as the family interface to hubitat for that reason.
Anyway, this method does require buying hubitat hub and setting it up and getting it running, as well as setting up the custom code on the smartthings side so that you can talk to each other. It’s more expensive and more work than just paying the sharptools license, but there are people who had a really big investment of time and energy in webcore who decided to go that direction. Choice is good. 
There are a couple different projects to do this. Hubithings Replica does not require a SmartThings/Aeotec hub if you don’t have one already. So it’s a good choice for people who are using webcore to manage Wi-Fi and cloud integration devices. And it offers a two-way integration.
Mira does require a smartthings/Aeotec hub, but then the good news is that the integration between the two hubs is fully local.
You could also use MQTT to communicate between the two, but that’s technically the most complex in this category, and probably only makes sense if you’re already running an MQTT broker. So I mention it for completeness, but I don’t think it sounds like something you yourself would be interested in.
Integration Solutions using MQTT
5) IFTTT does have webhook processing as part of its free tier
Normally, I would probably have listed this first, but the integration was recently rewritten to work with the new architecture and it was a little flaky to start with. So I’m not even 100% sure that it’s working right now, although the company does intend that it will.
Also, each free account is limited to 3 rules and I don’t know if that’s enough for your use case.
Technically, IFTTT on the free tier is, if anything, a little easier to use than webcore, and does not require a smartthings hub or another device. But it also is a much simpler rules engine, unless you pay for the subscription.
I think it’s definitely worth looking at, but, like I said, I don’t even know if it’s actually working right now.
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So there are several options for the new architecture, but free and easy may be harder to accomplish unless the Ifttt integration is working and can handle the details of what you want to do.