Database Technology for Large Scale Logging
Considering the volumes you are discussing moving it to the cloud isn't going to be magic fix. 

"There is no such thing as a free lunch"

A computer still has to process that volume, and that comes at a cost. If you are asking a 3rd party vendor to provide that theyre not just covering costs, but trying to make a profit too.

Changing to a different db platform can also have a lot of hidden costs. Even though licensing might be "free", there can be additional costs in maintenance, integration and compatibility. Even learning a new system is an overhead.

The spec you mention is not a big machine by server standards. I would be confident saying thats undersized for the volumes you describe.

What performance tuning have you done already? What are the bottlenecks? RAM/CPU/IO?

How much faster does it need to be? How fast is fast enough? Do you need a 20% improvement? 50% 100% You might easily get that with more RAM and some SSD's.

If you cant add more resources, is there anyway you can make the server do less work?

Archiving: Could you move old data off to a separate db. Or even better could you aggregate the data and purge the raw records?