How can I run old software on a new apple laptop and desktop?

You could install an older macOS to your current Mac (either to external or internal volume or partition) and dual-boot to that to run an old app. I sometimes boot my Mac mini 2018 to Mojave instead Sequoia to run old 32-bit apps like Lightroom, Photoshop, MPEG Streamclip, QuickTime Player Pro 7 etc.

Or dedicate and old Mac to run old apps. But beware that Adobe might have shut down license servers or otherwise made installing old app versions like Photoshop and Lightroom difficult.

How to download and install macOS - Apple Support

Create a bootable installer for macOS - Apple Support

Or use a virtual machine. Use an old compatible Mac to download macOS that can run the desired old app. My favorite old versions are macOS 10.11 El Capitan or macOS 10.13 High Sierra or maybe macOS 10.14 Mojave. OS X 10.6.3 Snow Leopard Server is the oldest I use but that might be hard to get nowadays (Retail version is NOT supported as a VM).

Then use VMware Fusion 25H2 (currently free), Intel Mac and Sequoia or Tahoe and install that old macOS as a virtual machine to run the old application. I don't know if Broadcom still offers older VMware versions for download. Other option is to use Parallels.

https://techdocs.broadcom.com/us/en/vmware-cis/desktop-hypervisors/fusion-pro/25H2/using-vmware-fusion/getting-started-with-vmware-fusion/system-requirements-for-vmware-fusion.html

Notice that new M series Macs can't run such old Intel macOS apps.

Re: “I also have Adobe software that is fully bought and paid for.”

Adobe discontinued one-time-purchase versions of most of their products so long ago that the installation and activation code is 32-bit and won’t even work on Intel-based Macs using Catalina or later.

There is no practical way to install old Intel-only versions of macOS onto Apple-Silicon-based Macs.

You will need to “bite the bullet” and “subscribe” to current Adobe software plans, or look into alternatives such as Affinity,

If by "new" you mean an Apple Silicon Mac it is unlikely you will be able to run "old" software on it, depending on how old the software is.

Without any specific details of the old system, if you want to run old versions of software on an old OS, you must usually use an older Mac to do so.

If you just purchased a new Mac with macOS 26.x Tahoe on it, then it cannot run an older OS unless the exact model you purchased existed prior to the release of Tahoe.....even then at best it could only be downgraded to macOS 15.x Sequoia. I'm not sure I've seen anyone confirm whether the M-series Macs can be downgraded to Sequoia under that very specific condition.

Besides, at some point support for Intel apps on an M-series Mac is expected to go away at some unknown point anyway so you will just be moving the line a bit. The writing is on the wall for those older no longer supported apps. You need to either keep an old Mac running, or find new alternatives which are compatible with your new M-series Mac and macOS 26.x+.