Regarding your violin, yes the action is usually higher on student instruments since they don't pay that much attention to creating the perfect surface for the fingerboard (bumps exist here and there) and a higher action prevents the string hitting those bumps and buzzing while you play. Also the fingerboard my have a way too deep scoop. But essentially that's why it requires more pressure to push down on the strings, since they're higher up. 

Let's discuss both cases So first.. let's talk about the violin playing.. technique / equipment..  

Regardless of what you're doing with the violin you need to make sure it's always a fairly comfortable position.. So make sure you have the proper shoulder rest.. adjusted to be the most comfortable it can get, chinrest.. etc.. When bowing try to be relaxed and only use your shoulder when you should. (a slight forward movement when playing near the frog.. and the up / down motion when crossing over to different strings) 

Getting a new violin is always nice But this isn't a problem with the violin, I mean the fact that you have to press harder IS a problem of the violin, but that won't really cause shoulder pain, especially in the bowing shoulder.. unless it makes you really really tense for some awkward reason... 

Now for the fitness stuff...

First of all, you didn't really go into detail as to what kind of exercises you do, but whenever anything involves repeatedly raising your arms while standing (shoulder press... lat raise.. stuff that brings your arm above your head) etc.. make sure you don't have an interior shoulder rotation. (your shoulders aren't more in front than your chest / collarbone). When that happens your shoulder joint isn't properly pulled back into its socket (that's why it's hard raising your arm overhead in the first place in that posture) and the repeated movements cause damage to your shoulder. So just keep in mind that you do all your exercises with proper form. Also try not to listen to bad advice you might get in a gym from other random people and research the exercises you're doing (in fact go right ahead and research the stuff I just told you! ha!) 

Anyway.. to relieve the pain stand with your back against a wall.. hold out your arms to the side with the back of your hand touching the wall, with a very slight bend in your elbows (just a tiny one, like the natural bend it gets when you turn your palm forward). Start lifting your arms slowly (same movement pattern as a jumping jack would be, only super-slow), while making sure that your shoulders, your elbows and the backs of your hands are all touching the wall. At some point near the top your elbows might come off the wall depending on how flexible you are, but that's okay   What this will do is pull your shoulders further back into their sockets and temporarily relieves all the pain. It's like a nice stretch repeat it like 4 times and you should be able to get back into playing. 

Keep us posted and good luck

Cheers,

Ferenc