Without a doubt don't even look at changing to NoSQL yet. Please be aware of the other considerations such as the benefits/downfalls of these systems. They are often JSON based, offer less ACID compliance, and are at various maturity levels; however for some data systems they are great.
NoSQL Benefits and Pitfalls (Cliff Notes):
They are often volume, throughput, or concurrency but often do that at the expense of ACID compliance on the DB Engine so you have to be very careful with your data governance. Often, you are fully responsible that your code has proper transaction controls, isolations, rollback capability, etc. You're not going to get that from the DB Engine anymore. Similar to the 2 DB engines of MySQL (MyISAM vs InnoDB) but on a much bigger level.
Systems such as Hadoop are good at processing petabytes upon petabytes, maybe scouring 10% of the internet every night and indexing it. This is not your scenario.
Others are more throughput based such as MongoDB which gives you a lot of parallel concurrency through dropping a lot of ACID compliance in the DB engine and putting the onus on the developer; which is great in some scenarios. However you only have 100 concurrent users so that is not even near an issue yet.
Google Elastic would be more useful for something that requires massive throughput. I mean 50,000 servers reporting into a cluster for near real time data for example. This requires tons of RAM usually, and this also is not your use case.
In your case, you're not even near the realm of big data when your entire data set can fit in $350 worth of RAM. You'll probably want to start with index analysis and what queries are the worst offenders that are often used. Try a free installation of a program like SolarWinds or many of the other ones. Install it on a new AWS machine, then delete it when you're done. Use it to profile your database server and see what is going on. Then you can diagnose the problem a lot easier and come up with solutions. Some are more complicated and require you to engage with a sales engineer while others can be installed themselves. I find these tools great for non SQL specialists.
See a bigger list here.