@Lance Lyons Thanks for reaching out. The logic app does have the Azure monitor logs connector and the available actions are listed here. As mentioned here the Azure Monitor Logs connector replaces the Azure Log Analytics connector and the Azure Application Insights connector now.
As there are only two actions available currently and both have the Query parameter where you specify your query you would like to run. Unfortunately, there is no way to run the saved query using this connector. You need to either update the query in the query parameter of your action or leverage the HTTP action in case if there is any API/endpoint that can perform for your operation.
Let me know if you have any queries or concerns.