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Choice D is correct. It's given that the shipment consists of $5$-pound boxes and $10$-pound boxes with a total weight of $220$ pounds. Let $x$ represent the number of $5$-pound boxes and $y$ represent the number of $10$-pound boxes in the shipment. Therefore, the equation $5x+10y=220$ represents this situation. It's given that there are $13$ $10$-pound boxes in the shipment. Substituting $13$ for $y$ in the equation $5x+10y=220$ yields $5x+10\left(13\right)=220$, or $5x+130=220$. Subtracting $130$ from both sides of this equation yields $5x=90$. Dividing both sides of this equation by $5$ yields $18$. Thus, there are $18$ $5$-pound boxes in the shipment.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect. This is the number of $10$-pound boxes in the shipment.