Welcome to the site.
There are (many) people on this site who have practised meditation themselves, but perhaps there's no-one who is very experienced as a teacher of meditation? It seems to me that this ("Body feels like crying") would be a question which an experienced meditation teacher will have heard before: so, one small bit of advice is, can you try to make contact with an experienced meditation teacher?
There are several forms of breathing (during "breathing meditation"): see for example Active breathing, passive breathing.
Anapanasati is most commonly practiced with attention centered on the breath, without any effort to change the breathing.
In the throat singing prevalent amongst the Buddhist monks of Tibet and Mongolia the long and slow outbreath during chanting is the core of the practice. The sound of the chant also serves to focus the mind in one-pointed concentration samadhi, while the sense of self dissolves as awareness becomes absorbed into a realm of pure sound.
In some Japanese Zen meditation, the emphasis is upon maintaining "strength in the abdominal area" (dantian or "tanden") and slow deep breathing during the long outbreath, again to assist the attainment of a mental state of one-pointed concentration. There is also a "bamboo method," during which time one inhales and exhales in punctuated bits, as if running one's hand along the stalk of a bamboo tree.
The above spends more words talking about chanting and the "bamboo method" than it does in the first paragraph about "passive breathing", but it might be right in saying that "passive" breathing is the way in which it's most commonly practised.
There's a book about Anapanasati, Unveiling the Secrets of Life:
a Manual for Serious Beginners by Buddhadasa Bikkhu, which has information about several (not all) techniques of breathing meditation, and which people recommend.
At the risk of going off-topic I'll also mention two non-Buddhist forms of breathing, from my own experience
Aerobic exercise (e.g. bicycling) -- I'm not sure how you can do "deep breathing" when you're not exercising, but exercise is recommended e.g. by (non-Buddhist) doctors as a way to help your body manage 'stress'. I mean, a bicycling fast enough that it begins to affect your breathing (and/or cycling up-hill), at e.g. 20 km/hour; or, various forms of running if you prefer (though I prefer cycling, myself, because it's easier on my joints). The experience (of well-being from being physically fitter) may make you more 'attached' to feeling physically fit (and perhaps monks don't recommend physical exercise, except perhaps "walking meditation"); nevertheless it (exercise) is meant to be one of the ways to cope with stress: see for example Exercising to relax from Harvard Medical School.
I've also sometimes enjoyed practising a "Tai Chi" form, which is also called a "moving meditation". There's a description of that (with links to videos) here. The caveats are that you need a reasonably good teacher; and a form with a reasonably low stance; and maybe a couple of years of lessons: because you need to learn the form before you can practice it, and you have to practice repeatedly (having already learned the movements) to learn to integrate the breathing with the movements; and learning a long form (e.g. 108 moves), which takes about 20 minutes to practice, might take a year (or more) to learn. Still I found that's it's good for breathing and posture (so, "physical well-being" again), and there's something to be said for its way of handling conflict.
Another (Buddhist) form of meditation which you might like to investigate is Metta Bhavana.
There are various ways of practicing metta-bhavana, the meditation on universal love. Three of the principal methods will be explained here.
In terms of insight about suffering, it might be worth considering "the Three Poisons" i.e. ignorance, desire, and aversion; for example:
- Aversion: I don't want to feel sad
- Desire: I want to feel happy
- Ignorance: I don't know why I feel sad
A common problem is, apparently, that if you think you're averse to something then you may try to avoid it. If you "feel like crying" and are averse to that feeling then there's a tendency to avoid inspecting that feeling, avoid thinking about that feeling. Instead maybe you need to be willing to investigate (accept, work out) such a feeling if it arises. I presume you'll find that such a feeling is transient (impermanent), because most feelings eventually are.