Restriction of Exact Cover by 3-sets
I didn't check if the Bangye's solution is correct/simpler, but a quick transformation from RESTRICTED X3C (the name is the same used by Gonzales)  to SINGLE OVERLAP RESTRICTED X3C (the name is invented) that should work is:


Replace each subset $C_j=\{x_1,x_2,x_3\}$ adding 6 new elements $z_{j,1},z_{j,2},...,z_{j,6}$ and 5 new three elements subsets $L_{j,1}=\{x_1,z_{j,1},z_{j,4}\}$,$L_{j,2}=\{x_2,z_{j,2},z_{j,5}\}$,$L_{j,3}=\{x_3,z_{j,3},z_{j,6}\}$,$L_{j,4}=\{z_{j,1},z_{j,2},z_{j,3}\}$,$L_{j,5} = \{ z_{j,4},z_{j,5},z_{j,6}\}$ .


like in the figure below (blue triples).
Informally, the three elements originally in $C_j$ are grouped and in order to include elements $z_{j,1},...,z_{j,6}$ the exact cover must include the group of triples $L_{j,1},L_{j,2},L_{j,3}$ OR the two triples $L_{j,4},L_{j,5}$, but not both.



At this point no pair of triples share more than one element and each element is included in exactly 3 triples; except elements $z_{j,1},...,z_{j,6}$ which are included only in two triples.


In order to fix this it's enough to add a duplicate of every element element $x_i \rightarrow x'_i, z_i \rightarrow z'_i$, a duplicate of each triple containing $x_i$ or $z_i$ elements using the corresponding duplicated elements (green triples in the figure below) and for each original triple $C_j$ add three new elements $t_{j,1},t_{j,2},t_{j,3}$ and 7 new 
dummy subsets with elements:
$D_{j,1} =\{ z_{j,2},z_{j,6},t_{j,1} \}$,
$D_{j,2} =\{ z_{j,3},z_{j,4},t_{j,2} \}$,
$D_{j,3} =\{ z_{j,1},z_{j,5},t_{j,3} \}$,
$D_{j,4} =\{ z_{j,2}',z_{j,6}',t_{j,2} \}$,
$D_{j,5} =\{ z_{j,3}',z_{j,4}',t_{j,3} \}$,
$D_{j,6} =\{ z_{j,1}',z_{j,5}',t_{j,1} \}$,
$D_{j,7} =\{ t_{j,1},t_{j,2},t_{j,3} \}$ (yellow triples in the figure below).


(note that the $z,z'$ elements of dummy triples $D_{j,1}, D_{j,2}, ..., D_{j,6}$ and elements $t$ of dummy triples $D_{j,4},D_{j,5},D_{j,6}$ are "shifted"  to avoid triples that share more than one element)



($\Rightarrow$) Suppose that $\bigcup_{j \in A \subseteq \{1,...,3n\}} C_j$ is an exact cover of the original RESTRICTED X3C instance. 
Then by construction:

$$\bigcup_{j \in A } ( L_{j,1} \cup  L_{j,2} \cup  L_{j,3} \cup L'_{j,1} \cup  L'_{j,2} \cup  L'_{j,3} \cup D_{j,7} ) \cup \bigcup_{j \notin A } (L_{j,4} \cup L_{j,5} \cup  L'_{j,4} \cup  L'_{j,5} \cup D_{j,7}) $$

is an exact cover of SINGLE OVERLAP RESTRICTED X3C.

($\Leftarrow$) Suppose that there exists an exact cover of the SINGLE OVERLAP RESTRICTED X3C instance.
Every original element $x_i$ must be included exactly once in the cover, but, as seen above, the only way to include an element $x_i$  is by choosing a group of triples $L_{j,1},L_{j,2},L_{j,3}$ that correspond to an original triple $C_j$ that contains $x_i$.
Furthermore  if $L_p, L_q, p \neq q$ are included in the exact cover we have $L_p \cap L_q = \emptyset$.
So the collection $L_{j,k}$ of subsets in the  SINGLE OVERLAP RESTRICTED X3C exact cover correspond to a valid cover $\bigcup C_j$ of the original RESTRICTED X3C instance.

The reduction can be done in polynomial time, so we can conclude that SINGLE OVERLAP RESTRICTED X3C is NP-complete.

Just note that a SINGLE OVERLAP RESTRICTED X3C instance built using the above reduction can contain two valid and distinct exact covers of the original RESTRICTED X3C problem, but we are sure that if only one exact cover exists, it can be "mirrored" to form a valid exact cover of SINGLE OVERLAP RESTRICTED X3C.

Let me know if you need a more formal proof for a paper.

Update 20/04/2020: I noticed that the previous figure had a small issue in the dummy triples and I fixed it.