Is it possible in Azure to clone one group of resources into another?
I tried to use "Export Template" in azure for this, but when I deployed this template, some resources did not load and I could not figure out why. Is there another way to do this? For example, something like through the azure cloud shell?

Most likely it will fail because of parameters that must be unique across Azure, for example DNS labels for Public IPs or App Services fully qualified domains, so you'll need to change those (and this can be a huge effort depending how many resources you have)
I found some alternatives, that I haven't tested to be honest but they look promising
RGCOPY (Resource Group COPY) is a tool that copies the most important resources of an Azure resource group (source RG) to a new resource group (target RG). It can copy a whole landscape consisting of many servers within a single Azure resource group to a new resource group.And it does support the following resources
The Export-AzResourceGroup cmdlet captures the specified resource group as a template and saves it to a JSON file.This can be useful in scenarios where you have already created some resources in your resource group, and then want to leverage the benefits of using template backed deployments.This cmdlet gives you an easy start by generating the template for your existing resources in the resource group. There might be some cases where this cmdlet fails to generate some parts of the template. Warning messages will inform you of the resources that failed. The template will still be generated for the parts that were successfulThis is similar to the Export Template that you were using, and there are some limitations
Copy Resource Groups
can be done, but it requires considerable time and effort to manually recreate the network resources, copy the virtual hard disks and attach to newly created virtual machines. To eliminate the manual effort I created a Powershell scriptHere is his repository https://github.com/JeffBow/AzurePowerShell, and you will find two scripts to give a shot
References: