What pet/pets are you using, and how are they specced?
Are you keeping your pet/pets at or close to your level?
How are you specced?
I have soloed most of the content except for dungeons and some of the harder elite quests. I am still going through the endgame areas, as I am here for the story/experience and not to power my way to endgame. I can offer some advice, though I am by no means an expert. I am leveled as Beast Master (BM) only speccing into Marksman (MM) after I had everything I wanted from BM. Survivalist did not really have anything in it that appealed to me that was feasible to get. Your mileage may vary, depending on how you have chosen to spec of course.
As a general rule of thumb do not try to engage more than two comparable (yellow) leveled enemies at a time, unless you are very familiar with them and know you can take the group. Even then I would only try to engage three enemies max. Never try to engage multiple elites unless they are grey and you outlevel them by a decent margin, and there again do not surpass three.
Take your time between fights to freshen up. You may be trying to rush things too quickly. The spirit bonus from “Well Fed” helps cut down on down time between fights, and helps when the five-second rule kicks in during the fight. Your pet also seems to have a fairly decent out of combat regen compared to you. Consider letting it regenerate completely between fights.
Keep your pet well fed. A happy pet (green face) does 25% more damage, which in turn helps burn down the mob quicker, which means less healing overall.
Be conservative with your mana. I try to not use too many active abilities in a non-elite fight. Personally, I tend to place a trap as a fallback, then open with “Hunters Mark”, followed by letting the pet attack for a second or two to build up threat so that I do not pull aggro, followed by a sting of what I feel is the appropriate type that I may reapply as needed throughout, mana permitting. Healing as needed. Always keep enough mana for an emergency pet healing, and later an emergency pet healing + FD. When I finally got “Aimed Shot”, I added it in after the initial sting. I generally only “Aimed Shot” once per enemy. Most of your damage will come from “Auto Shots” unless you have lots of +Intelligence gear. This also has the benefit of allowing the five-second rule to kick in and your mana to start regenerating mid fight. I do not recommend using “Arcane Shot” at all, unless you are facing a very high armor enemy. It simply is not mana efficient and does not benefit from “Hunters Mark” or “Aspect of the Hawk”, and I find “Serpent Sting” to provide more damage per mana spent (assuming the enemy is not immune).
Be extremely conservative with “Multi Shot”, and when you get it, “Volley”. These are very mana hungry, and generally do not pay off unless the mob is very weak (silithid swarmer spawns) or much lower level than you. Without a player tank, you are generally just going to draw aggro from all the enemies your pet is not directly attacking. That said, as others have pointed out, you can use this to your advantage.
Do not try heal your pet when you are taking damage, unless absolutely necessary, or the enemy attack is a spell. Caster pushback will shorten your channeled heal making you waste mana.
Try to focus on the same enemy your pet is attacking to kill it faster, and thus remove it as source of damage. However, pulling off one mob to attack you leaving two for the pet to deal with is viable. Depending on the enemy in question, you should be able to take the blows while fighting with your pet from range. If not, melee your attacker while the pet fights its enemies.
When you do get the “Feign Death” I would save it as a panic button for when things go south. The FD + Pet Passive + Frost Trap is not reliable in my opinion and is apparently buggy (though I do not know how it behaved in Vanilla). I would much rather run away a bit, trigger FD, and then let them kill my pet. You have to resurrect your pet, feed it to make up for the 350 death happiness loss, and lose any progress when the mobs health resets, but it is better than dying yourself. You can use this tactic to wear down a pack slowly through attrition, one member at a time, if needed.
Stings can be a situational too but ones that should not be ignored when soloing with your pet:
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“Serpent Sting” will generally do more damage than an individual shot, making it ammo efficient at the cost of mana if the fight last long enough. Depending on your gear, it may do two to three autoshots worth of damage during its duration. This may seem somewhat counterintuitive if you are having mana issues, but I find that it helps as the enemy gets burnt down quicker with pet attacks + DoT + my auto attacks, which means less damage taken and need to heal mid fight. I will note that I craft ammo when at all possible so I am more ammo usage conscious than some, which has no doubt colored my view of this ability as well. I would say try it at the very least. It may not work with your style of play.
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“Scorpid Sting” is really only useful for hard hitting enemies, as it will lower their damage output slightly. Hopefully, it will bring it down to a manageable amount. I do not know if the decreased Agility effects the enemies armor/defense in any way. I have not tried this with the talent that decreases Stamina from the MM tree. That might make it slightly better. Someone else would have to comment.
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“Viper Sting” is sadly not useful in PvE content 90% of the time in my experience. This is unfortunate, as I would have loved a way to deal with casters/healers other than focusing on them first. Most enemies have more mana, or higher mana regen, than this will drain. Maybe it might do something for a hybrid enemy, but pure casters are unphased by this. Do not even bother trying to use it on an elite enemy. It may have PvP utility, which depending on your server may be worth considering. I can tell you as a primarily solo Hunter, we are very vulnerable to attack from others when we are engaging mobs.
Another tactic that I find useful that you may one to try once you have the abilities is as follows: Lay an “Immolation Trap” between the enemy and myself, back up a ways (max range ideally and pull the first enemy with a “Serpent Sting” into the trap, and then once the enemy is about halfway between its starting position send my pet to attack it. This allows for letting it take some damage before it can do any of its own, and leads to it taking pet attack + my attacks + two DoTs simultaneously. This is good for wearing down tougher enemies like Ogre Maulers/Lords that also hit hard.
Alternatively, you can use a similar tactic with “Frost Trap”, just do not start with a sting and have your pet and you attack a second enemy. When the trap wears off, have the pet momentarily switch to the unfrozen enemy to draw its aggro and then switch back to the previous enemy. This allows you to take an enemy out of play for a time, hopefully, killing or severely damaging another during that window.
Finally, when going solo do not be afraid to leave and come back later when you have better gear/abilities.
Feel free to ask any questions.