First, I want to make sure you're not making the same mistakes that most people make when it comes to calves. Doing calf specific work won't necessarily make your calves bigger, unless your chemically enhanced via steroids. When you're a natural lifter, most of your muscle growth (except for when you first start lifting weights for a few months) will happen on a systematic basis. What this means in a nutshell, is something I'm sure you've heard before...big compound movements.
Big compound movements, release more hormones, invoke more protein synthesis, and improve neural efficiency as they recruit more muscles, and energy.
In terms of calves, think about this for a second, when's the last time that you saw someone who could deadlift or squat over 500lbs and had small calves? Probably very rarely...if at all.
Now think about how many people you've seen at the gym with a massive upper body with basically twig calves? I know, I've seen a lot. This is due to not doing lower body compound movements, or just not doing them enough... for example squatting once a week may not even be enough...think about it! Let's say you do bench one day, shoulders another, and arms another...you are engaging your triceps THREE times a week as opposed to training your legs only once, I'm not saying you do this, I'm just trying to show you a picture. Now, when I had small calves, the way I overcame it was by focusing on working my legs more frequently in general, i.e twice a week, and of course adding some calf work at the end of these workouts. Because remember, you can't have big calves without somewhat of a big squat or deadlift...unless you have some great genetics, or are old (old ppl have massive calves usually)
So, to work calves, I would suggest, first do a lot of squats and deadlifts, as your lifts go up in these two movements you'll be surprised how much your calves grow. Remember, when you're natural you're more likely to grow systematically, so you need big movements like these. Particularly, a GREAT exercise for calves that most people don't seem to know about is the stiff legged deadlift, or romanian deadlift. I'm not gonna go over the form here as I'm sure you can look up the details online. But it is VERY good for your calves and a compound movement so even better. If you don't believe me, stand up straight with your legs straight, knees slightly bent, and try to touch your toes...feel the stretch in your hamstrings and calves?
Once you get the big movements out of the way like squats and deadlifts, THEN you should be focusing on some direct calf work if they're your weakness. I will tell you about a very nice trick for working calves at any gym. First, the machine that you have is actually great, and over time as you keep doing it, your shoulders will adapt and so you shouldn't really stop...maybe just go a bit lighter and do higher reps until u can move up. Now, your calves do not need a lot of weight in order to be stimulated...they are one of the few muscle groups that respond well to high rep work, such as forearms and traps. (This is because you use your forearms and calves EVERYDAY in any activity so they need slightly different stimulus in order to invoke growth). Furthermore, something most people don't know is, that your calves mostly function as stabilizers! Now given these two assumptions (they're more or less facts by now, but I'm sure some people won't agree so...), you can do A LOT of different movements to invoke calf growth.
My favourite would be to lay down two 5 or 10 or 25 lbs plates on the ground. Grab two dumbbells , one in each hand, (weight about 20-60lbs ...really depends on you). To begin, place the balls of your shoes, on the weights on the ground so that your heel is still on the ground but your upper feet are on an angle (similar to how you would start on a calf raise machine), then with the two dumbbells by your side, slowly do a calf raise, going ALL the way up , pausing at the top for 1-2 seconds, squeezing the muscle, and going ALL the way down and feeling the stretch . Do this for about 2-5 sets of 10-25 reps...increase the weight as you progress. You can also do this on a smith machine with barbell on your upper traps, but I would really suggest away from this, since like I said, your calves are stabilizers, so stay away from machines until the very end where you can overload the muscles with more weight. I.e, you can just get in a normal squat position with the bar on your back with more or less the same weight you squat with, but just do the weight elevated calf raises instead. At the end, go find a leg press machine, horizontal or vertical doesn't matter...push the weight up with your legs so that your legs are almost straight, but NOT fully locked as this will destroy your knees, keep the weight on the balls of your feet, i.e from half way up your shoes, keeping your heels not touching the platform at all, and do calf presses there for a few sets of 10-20 reps. Search up leg press calf raises for more information if you want. But after squats or/and deadlifts, you shouldn't need more than 1-2 movements for your calves.
Moreover, other than your basic calf raising movements, another great way to stimulate your calves would be to do explosive plyometric jumps. Find a hip-level box, step on it, then "walk" (NOT JUMP) off of it so you impactfully land on the balls of your feet and IMMEDIATELY jump up as high as you can, focusing on pushing and following through with your calves...again you should look this exercise up online as well for more help.
As a final note, if you REALLY have tried everything and your calves just won't grow, it would be a good idea to switch things up and work out your calves FIRST in the workout. Exercise priority (order of exercises performed in a workout) actually matters quite a lot in the long run and will benefit lagging muscles. So go to the gym, start with 1-2 of the calf movements I mentioned above (there's tons online as well, way more than any of us could list), after these your calves will be tired, so that when you perform big movements like a squat or a deadlift, your calves will be exhausted and thus will have to work twice as hard in order to keep up...effectively making the exercise more biased towards training calves.
Hope that helps.