Next-gen projects have been renamed to Team-managed projects. However, you won't really be able to do "swimlanes" as such with those. I would suggest you create a Jira Software Company-managed project.
So, click on Create project like you did, then select Jira Software.
Then click Kanban (or Scrum if you are using Sprints).
Then select Use Template.
Then select Company-managed.
And follow the steps from there.
As John said, Next-gen projects were rename to Team-managed projects.
It is important to understand the difference between company-managed and team-managed projects:
Set up and maintained by anyone on the team, team-managed projects are ideal for autonomous teams who want to control their own working processes and practices in a self-contained space. Team-managed projects have simpler project configuration and give project admins more control over set up without involving a Jira admin (unlike company-managed projects, where one is required to configure schemes and screens).
Set up and maintained by Jira admins, company-managed projects are the best choice for teams who want to work with other teams across many projects in a standard way, such as sharing a workflow. Company-managed projects require a Jira admin to configure screens and schemes that projects are based on. Hence, company-managed projects have greater complexity in project configuration, but also the ability to standardize workflows across projects that team-managed projects don’t have.
I strongly recommend checking the documentation below to understand better the difference between the two projects:
Please give it a try and let me know how it goes.