Hi KidRandom and welcome to The Mattress Underground 
Hello, I recently started renting a new apartment and need a new mattress. I describe myself as a side-sleeper but I often wake up on my back. I’d say I’m primarily a side-sleeper but sometimes sleep on my back. I’m 120 lbs. I don’t seem to like firm mattresses
Any mattress is unique to those sleeping on it- based on their Stats (height, BMI, sleeping position(s) and any underlying health conditions), and their PPP ( Posture and alignment, Pressure relief, and Personal preferences). You may want to take a look at the Mattress Shopping Tutorial, as well as the Mattress Specifications You Need To Know and the Mattress Durability Guidelines, to get an idea of wheat to look for in a potential mattress. Next, you need to decide on what type of mattresses you feel comfortable on - a visit to a local showroom can allow you to ‘check off’ mattresses that don’t meet your criteria.
I’ve been staying with someone sleeping on their Casper Hybrid and it is way too firm for me and it’s been causing me shoulder pain. I tend to switch sides at night too and this mattress (I guess because of the memory foam) makes doing so difficult. I prefer to sink into a mattress and not feel like I’m completely on top of it.
I’ve been looking into softer/plush mattresses, especially ones from Helix (in particular the Sunset and Moonlight), but also looking at other brands. What would you recommend I look into?
With your lower BMI, if there are no complications in your Stats and PPP, you have plenty of options. Hybrid? Latex, micro oil layer? You are looking for a plush mattress -as long as you feel comfortable and your alignment is good (when laying on your side your back should look straight as it does looking at your back when standing), it will be a matter of preference.
Memory foam is popular with many consumers, but its’also know for sleeping ‘hot’…and the 2 Helix mattresses you mention, as they are made with proprietary foam formulations, are not transparent about either the density nor composition of their components. Excessive amounts of inexpensive and lower density foam can break down prematurely, leaving ruts or divots that can cause aches, pain and spinal misalignment, for example.
Many of the Trusted Members of the site carry Various hybrid, foam, latex and other compatible sleep solutions. These all take pride in the quality of their offerings, total transparency on their components and excellent customer service. If you can provide some info in your Stats and PPP and mattress history, they can help you find your next great mattress. If you want to provide your Stats and PPP and a few mattresses you are looking at, we can certainly comment on how they may work for you as well.
~ Basilio