This is current to me, but I'm going to have to speak in metaphor a little. It's also personal and subjective, so disclaimer. I'm also not a medical expert. I'm sharing my experience so you may understand how the road has been walked by others
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If depression is the equivalent of drowning, antidepressants are the reassurance of concrete under your feet in the swimming pool. It tells your brain 'this is not the ocean, this is a swimming pool, and the maximum depth is five feet deep. You can stand.'
It won't pull the cold from your bones, or the weight of the water on your chest. But you know the pool is five feet deep. You know you can reach your feet down into its depths and you'll find footing. And perhaps another may prefer the joy and exhilaration of a bottomless depths, to tread waves under a golden sun, knowing there's a boat/raft/pier/floaty/whatever to climb upon and bake away the water's chill. But say you are a drowning man, or you cannot enjoy the water for that terror of oblivion, and you need to know there is solid footing below you. You need to know you can reach the edge of the pool and climb out. This is analogous to 'emotional flattening.' You still have your emotions; you feel them, but you have leverage to push against the bottom and navigate when you feel lost.
It gives you purchase to guide yourself of your own strength. You will still need strength. Sometimes the pill is enough, sometimes you may still need therapy, or habit changes, or supporting friends/family. You also need to be aware that it may increase suicidal ideation. Don't misplace your strength in this way.
It's oil on the gears in your mind- gears locked together by exhaustion and sorrow now slip and turn, and it becomes easier to grasp the fuller machinations of the self instead of being crushed between friction points. Now, you may sometimes find the teeth of the gears don't find bite when you're cranking it, and in that sense you may lose your highest highs in addition to your lowest lows, and that may really, really suck depending on how you value it. It's a pyrrhic victory, but may be the victory you need.
It may tank your tolerance for other recreational substances (alcohol, weed, etc). I can no longer safely drive on two drinks. It makes me a real cheap date!
Your antidepressant may also help with anxiety; it may not. This is currently the point I'm struggling on, which leads to the next point - it may take a few different antidepressants to find one that fits right for you, and it's a long and uncomfortable journey. Usually they take 4-6 weeks before you start to estimate efficacy, and onboarding can be unpleasant- you may have bouts of insomnia, headaches, brain fog, increased appetite, decreased appetite, anxiety, depersonalization, derealization, nausea, diarrhea. Much of it should abate after the first few weeks. If it doesn't, or you just don't like how it makes you feel, or your depression isn't helpfully managed by it, then you may have to change the dose and go through another bout of adjustment, or try a different one entirely - and that means getting off a substance your body now feels it's dependent on. These are your withdrawal symptoms, and they can also be unpleasant.
(as an aside, I know there's a fair cultural weight and stigma to the idea of chemical dependence/addition/withdrawal, and it's crucial to separate that from your antidepressants. you're not joyriding on happy pills, you are fixing a dysfunction in your brain. you are not an addict, you are not a problem, you are not a flawed human being. you are negotiating your greater suffering with smaller or temporary sufferings- and the withdrawals are neither small nor brief! neither is your depression! you need to shush that part of your brain that says 'is it really that bad, or am i faking my depression/anxiety?')
Your withdrawals may also take several weeks. When I came off escitalopram (lexapro), I would get a brief (milliseconds), surge of vertigo when I'd turn my head or shift my eyes quickly - the 'brain zaps' you may hear about. That took about two months to go away. I also had some emotional regulation issues (granted, I was also in a bad spot - taking care of a house, two dogs, and a chronically ill partner who doesn't love you while far, far from friends and family is bad for one's mental health), which got better, but it was uncomfortable to discover a rage inside me I don't normally have. It's important to give yourself grace and know that you're shaping a better you, and that the temporary instability isn't you.
And maybe you need your antidepressant for the forseeable future. Maybe one day you won't. Maybe you decide that the whole ordeal isn't worth it. This goes in two ways - a) you can stop your antidepressant if it's truly not what you want (but do so with guidance of your doctor!). You're not forced to take it on threat of organ failure or anything; it's not a permanent change to your person. However, b) If it DOES help, it may be foolish to think 'well, I sure do feel better, what do I need this for any more?' and then you become dysfunctional again. It's a learning process, and possibly humbling if you're not ready to assess your dependency.
It's not easy. None of it's easy. But that's why we're here, isn't it?
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The following is highly personal- I share how they made me feel for context, not to suggest they could make you feel the same.
Speaking for SSRIs - I'm familiar with escitalopram and sertraline. To generalize, they feel/felt a little steely gray/cold (abstract), objective, and capable. It's a 'swallow your worries and just get it done' sort of feeling. Sertraline has markedly improved my baseline emotional level. They have not stopped the feeling of my nerves catching fire and brain screaming at me that I'm being hunted for sport. Escitalopram did not stop the one full-blown panic attack I had last year. Speaking personally, sertraline is being ruinous to my bowels. They make me clench my jaw, and I get a sort of fatigue that makes me yawn a fair bit without actually making me feel sleepy or exhausted.
I've also been on bupropion- I didn't feel or get much from it except maybe a quiet softness/confidence in a 200mg standard release level, but once I bumped up to the 300mg extended release, I felt like the cathrode ray emitter end of a CRT TV, like reality was something that happened just outside of me. It squashed my anxiety down flat, but it took my desire to do, well, anything, with it. I'm not on that one any more.
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Sorry, I know it's a lot of text, and more than what you asked for with 'what it feels like.' It's a lot to commit to and it's a long process. You're not alone.