I disagree with the previous answer. The answer is not 'No;' it's 'It's not that simple.'
The very concept of 'full custody' is not quite right. There are two related issues to child custody: legal custody, and physical custody, or parenting time.
Legal custody refers to decision-making authority over a child. A parent with legal custody decides where a child goes to school and what religion they observe, makes medical decisions for the child, and so on. Parents can agree to joint legal custody, sharing this authority, but a court will not order joint custody unless the parents agree to it. When we talk about "full custody," this is usually what we mean.
Physical custody, or parenting time, refers to where a child lives. Each custody case ends with a parenting plan describing where the child or children stay. These plans can be as specific or general as the parents need. The parents are free to deviate from the plan if they agree to do so, but the plan is there for when they can't agree.
Usually, a parent who has legal custody of a child also has more parenting time, but this is not required. Parents can share equal parenting time while one has legal custody; or parents can share joint legal custody while one of them has physical custody most of the time.
But child support payments are based on the plan for physical custody. Child support is calculated based on the amount of time (measured in the number of overnight stays) that each parent has with the child each year, and each parent's income. A parent who has 51% or more of each year with the child receives support; the parent who has less pays support. If two parents have exactly equal parenting time, then the parent who earns more money may pay support anyway. This page contains an application for computing the child support amounts presumed correct under Oregon law: https://justice.oregon.gov/guidelines/
Who has legal custody doesn't directly affect child support. It's the parenting time schedule that alters your support obligation. So yes, you do have a financial incentive to maximize the amount of time you spend with your child each year.
Courts make decisions about legal and physical custody according to the best interests of the child. "I don't want to pay child support" is not a reason a court will consider to be good; in fact, if anyone ever heard you say that, it would be strong evidence against you getting more custody rights. Your ex could say that you don't really care about your child, you just care about paying less support. If you want a court to order a given schedule, and its resulting child support obligation, you have to demonstrate why a given schedule was in the child's best interests.
You should be aware, too, that if you owe child support, then you will not be able to evade your obligation by working "under the table." Even if you avoid getting a job with a paycheck so that you don't get your wages garnished for support, you're still accruing a support obligation every month. That debt will continue to mount, and will accrue interest at 9% per year if unpaid. It cannot be discharged in bankruptcy, and will not go away even once your child turns 18. Refusing to pay child support can have serious consequences. It can eventually get your driver's license suspended, and can even get you put in jail if it can be demonstrated that you evaded your obligation intentionally. What you need to do, it sounds like, is to modify your support obligation to reflect your actual income. If you have 50% parenting time, then you shouldn't be paying that much in support.
You should consult with an attorney if you're contemplating legal action. You can call the Oregon State Bar for a free referral at 503-684-3763.