Feedback also from my side what I learned from projects.
1. Prioritize the test cases
In my past projects we prioritized the test cases. We used HP ALM and there we had also couple of test cases and it was impossible to execute the all. So what we did is just to prioritize the test cases e.g. Critical, Very High, High, Medium, Low - same as you would do it for defects. All created test cases were based on release policy which we also created. This helped just to focus on the most important test cases
2. Involve others in testing scope e.g. Product Owners and shift some test cases to them
Involve the otherst in the test team. Testing doesn't mean that you just test alone with your testing mates. It means that you can also involve other team members. In our project we hence involved also the Product Owner for testing. And not just at the end (e.g. when deployment is done and UAT should be done via business department and Product Owner). In Janets book I found the hint that UAT is done mostly after post deployment. and that tis does not mean that it has to be left to the end game (Janet Gregory "More Agile Testing" page 202). So when testing a product of course you can involve other stakeholders to test a product. In our case our Product Owners helped to test some test cases! By the way, you also can ask UX department to support you. We asked also our UX department to support us to execute some test cases and we got also valuable feedback from them.
Organize somehow Mob testing and act as test coordinator who throws test cases to them!
You can also try to do a Mob testing session. Just give them some test cases (which are easily understandable) and some instructions (and prepare e.g. test data) and just let them run the test cases! We did this when we invited some stakeholders from other department just to support us! This worked and the inputs was very helpful for us (because we also got feedback for new Change requests). The good thing was, that we just send one from our testanalysts there who was responsible for four-five groups of two persons. Somehow this is like a accelerator for proceeding with test cases.
Ask the business department for support
We asked also the business deparmtent for support. Generally the business department only takes action when the UAT should be done. But we asked them to support us, and we gave the reason that the help would be valuable in order to deliver a (testing) stable product.
Use a logging tool for exploratory testing
We reduced also our amount of time to create test cases when we made the decision to use a capture-replay tool. In our case it was tricentis/quasymphony. During test execution this tool creates test cases and during that time we did somehow exploratory testing. Means execution and creation of test cases. So this reduced somehow time for creating concrete test cases.
You see, there are a lot of ideas. But the most important one is that testing is a whole team approach! So you can invite all relevant stakeholders to support you - and based from our experience mostly were willing to support us.