If we follow this line of thinking then we must conclude that the watch wasn't designed either, it's also a product of evolution because humans are a product of evolution.
This is just making an equivocation of 'design'. You can have 'design' in the general sense of organization and order, both natural and intentional (e.g. a snowflake or a watch),
or you can have 'design' as solely the product of intent - artefacts of creatures capable of planning and modeling (not just humans - there are more creatures capable of this than you might think). When you equivocate the two, you get the kind of fallacious argument you suggested.
An alternative way of thinking of it is that there are two kinds of design - design by stepwise refinement and design by deduction. Design by deduction is the process of deducing the design from the requirements of the goal, something only possible for a creature capable of abstract thought. Many human designs typically start with a design by deduction and are then advanced by stepwise refinement (e.g. watches).
However, design by deduction isn't an essential start point, because existing objects can be used as a basis for stepwise refinement - i.e. making a series of modifications to a found object until it is suitable to achieve some goal (e.g. bending a wire coat-hanger until you can open a car door through the side window).
But you don't always need to modify the same object during stepwise refinement - sometimes selection by trial and error from a population of varying objects can eventually result in an effective design - consider someone on a stony beach trying to skip stones for the first time, who throws many stones before finding that a flatter one skips better than a round one, then continues to refine his selection of flatter stones until he finds the ideal shape for skipping. In this case, the goal of the selection process is set by the stone thrower who drives it. Is his last selection, the best skipping stone, a design? Kind-of, I guess it's a matter of opinion.
With simple life, the selection process itself is the driver for the stepwise refinement from a population of variants - there is no explicit goal; but because only the variants that survive can reproduce to continue the process, it gives the appearance of having survival as an implicit goal, and gives the appearance of design for survival - although these are just anthropomorphisms.
To continue the beach analogy, the random mixing and interactions of organic molecules prior to the first replicating molecules appearing (i.e. abiogenesis) can be seen as analogous to the man on the beach randomly throwing stones into the water until one happens to skip, and the persistence of the few replicators best able to replicate in their environment as analogous to the subsequent selection of better skipping stones.
