Whilst there are high-quality, paid-for enterprise architecture applications in the market, with the increasing move to SaaS and dangers of vendor lock-in, it's prudent to have an alternative. Does anyone have experiences (good or bad) to share after moving from one of the market leaders to open source alternatives Essentials and/or Archi? https://enterprise-architecture.org/products/essential-open-source/ https://www.archimatetool.com/

The two tools are the only real open source ones, i'm aware on either. But the focus differs a lot: Archi is the official implementation of ArchiMate of The Open Group. It has the potential to model not just the EA part, but a lot of Software Architecture as well. To the lowest level possible, it is even comparable to UML. My personal bed is, this is the preferred tool by Software Architects or Developers. It stores the information in small grain files on the local file system and can be integrated with Git repos for collaboration. 
Archi is lacking any kind of reporting and dashboarding. All models are on separate graphical canvas.

If you are striving to portfolio based approaches for Application Portfolios, geographical distribution, technology spread, then Essential Architecture is by far more convenient, because it is offering ~ 80 out of the box reports, that can be combined to Dashboards for different audiences. Data maintenance can be done by batch uploads to a certain extend in Essential Viewer using the import & export tool. The data maintenance on individual entities is done with an ontology editor (Protegé). That can be installed with a shared database and commenting functions, but the data entry in the Open Source version requires intensive training and understanding of the underlying meta model. If you extend the metamodel in the ontologies, you can even add new attributes, objects and add them to reports and dashboards by configuration, but endanger upgrade compatibility. There are good reasons, that the Cloud version (paid) has online editors in the web browser that avoid the ontology editing.

If your need is mainly to explain and present, Essential Viewer might be the right tool.
If your need is mainly to model from a single application perspective, ArchiMate might be the right tool.
Both tools require experts for data entry, because both meta models are quite complex and you need to know, what data to enter in which element of the meta model.

These are my two cents on your question.

I'm very happy with the Archimate Tool. I use it in my private time a lot, and even at work where we do have the more complex tools/suites in place, it is something I regularly fall back to for quick evaluations or modelling needs.

While in our company we moved from the Archimate Tool to Sparx Enterprise, it wasn't because of disappointments in the tool. Most of the architects were happy to use Archimate Tool. The issue we had was that we needed to move towards more centralized repository-based working, as well as additional design expectations (such as data modelling). While Archimate Tool does have plug-in capabilities to work with central repositories, the application owners here found that the Sparx Enterprise tool would be a better fit.

The Archimate Tool is simple, fast. When you work on architectural modelling individually, it's a perfect tool. It supports the model exchange format, so you can export your ArchiMate models from one tool to another. And through the Jasper-based reporting, you can easily customize report generation.