I think I can see an example using decomposition theory. I am mostly using the notation from [the wikipedia article on the Whitehead manifold][1]. I am not sure yet, whether it is an example for $n=3$ or $n=4$. Take $K_1=S^1\times D^2$ and $K_2=(S^1\times D^2) / W$ where $W=\bigcap_n W_i$ is the intersection of a sequence of nested solid tori such that each $W_{i+1}$ is embedded in $W_i$ as a Whitehead-link and we use the standard framing (although I am  not sure whether this makes a difference for the argument). 

Certainly $S^1\times D^2\times \mathbb{R}$ can be embedded into $\mathbb{R}^4$. Thus knowing that it is homeomorphic to $K_1\times \mathbb{R}$ would show that $K_1$ embeds into $\mathbb{R}^4$. I don't know whether it embeds into $\mathbb{R}^3$.

I believe that Bing shrinking can be used to show this and that I heard about this example in a lecture quite some time ago. 
I looked it up in Theorem 7.5 in "Behrens, Stefan, Boldizsár Kalmár, Min Hoon Kim, Mark Powell, and Arunima Ray, eds. The disc embedding theorem. Oxford University Press, 2021.", which again gives 
Andrews, J. J., and Leonard Rubin. "Some spaces whose product with E^1 is E^4." (1965): 675-677.
as a reference

  [1]: https://en.wikipedia.org/wiki/Whitehead_manifold