If, within the framework of modern society, individuals depend on employment for survival, then how can one survive without breaking either their moral aversion to wage slavery or their moral aversion to moving their financial burdens onto others?
People who work for a wage aren't slaves and slavery is not an apt metaphor. A person must have food, shelter etc. to survive. Nobody is obliged to provide you with these goods. If you maintain otherwise, then you are saying the person who can provide those goods should be forced to work to provide you with them: he is your slave. The person who can provide those goods should do so only if he has been persuaded to do so by argument, or by receiving goods he wants in exchange.
One of the options for obtaining goods that you can exchange for food and shelter is that you can agree to do stuff another person asks you to do in exchange for money to be paid in a short time - a wage. In order to pay a wage somebody has to gather the resources necessary to pay that wage and do without them for the period required to produce something that can be sold.
Some people say the employee and employer have unequal bargaining power and that this is unfair in some way. A person looking for wage work can consider an offer and negotiate or reject the offer. He need not accept the first figure that the employer proposes. There are books on salary negotiation if you don't think that's an option. If an employee wasn't providing value nobody would want to hire him. If the employee fails to think about this issue and look for information about how to use it, that's the employee's responsibility. The employer is not obligated to do this for the employee. There are other options, like starting a small business, or a large business. Promoting the idea that employers have power over employees is a vicious lie. That lie demonises employers and claims that employees are not moral agents who can make choices about what they value and how they want to obtain it.
A commenter below complains about employers treating employees as a means to an end. An employer does not treat an employee like a cog in a machine because the employee can resign. The employer would then have to find a replacement. However, the employer won't pay someone to watch cat videos on youtube. The employer will demand and should demand that the employee will do something of value to him to get paid.
The current system has many defects, like lots of burdensome govt regulation that stop people from starting some kinds of businesses, but that's govt slavery not wage slavery. If you complain about wage slavery, then you have a beef with the laws of physics since they don't require cooked roast chickens to fly into your mouth with no effort on your part.
It would be a good idea to learn some political economy. See "Time will run back" by Hazlitt for an explanation of the relevant economic issues that you might find interesting:
A short explanation of the moral issues can be found in "The virtue of selfishness" by Ayn Rand.