For your first problem, some applications automatically run when the computer booted up. I doubt the service may be locked by the application which was trying to establish the internet connection, when you were trying to start EASUS Partition Master. And I suggest you wait a few minutes, then try the operation again.
If there is not enough unused space in the OS partition, you could transfer some unimportant or inessential data to other partitions temporarily and get the enough unused space for resizing the OS partition. Or change the location of the page file for virtual memory to other partitions temporarily, if you set it in your OS partition.
After you fixed the first problem, did you run the “Check Partition” function in the product ? If not, please have a try. And please clarify the meaning of “trying to move the large hard disk that has plenty of space.”
And to extend the OS partition, please refer to this link for more information.
In order to ensure a smooth and successful operation process and data protection, please check this link before operating the partitions.
To further analyze your problem, please post the screenshot of your Windows Disk Management here.