How to we prove if a right linear language is ambiguous?
For simplicity, let me assume that the only allowed rules are of the form $A \to aB$ and $A \to \epsilon$. Denote the set of nonterminals by $V$, where $S \in V$ is the starting symbol.
Construct a new right linear grammar with nonterminals $V \times V \times \{0,1\}$. The new grammar will simulate two different production sequences for each word, keeping track whether the two actually differ.
The starting symbol is $(S,S,0)$, and the rules are:

For every pair of original rules $A \to aB$ and $C \to aD$ we add the rule $(A,C,1) \to a(B,D,1)$. (Possibly some of $A,B,C,D$ are equal.)

For every pair of original rules $A \to aB$ and $A \to aC$ such that $B \neq C$ we add the rule $(A,A,0) \to a(B,C,1)$. (Possibly $A=B$ or $A=C$.)

For every pair of original rules $A \to \epsilon$ and $B \to \epsilon$ we add the rule $(A,B,1) \to \epsilon$. (Possibly $A=B$.)


A word $w$ has two different derivations in the original grammar iff it can be generated by the new one.
For example, in your sample grammar the word $ba$ has two different derivations, $S \to bA \to baB \to ba$ and $S \to bA \to baD \to ba$, and this is reflected in the following derivation in the new grammar:
$$
(S,S,0) \to b(A,A,0) \to ba(B,D,1) \to ba.
$$