How to draw Turing machine for multiplying a number by 2 in base 10
To elaborate on the method described by Yuval in the comment, first, construct a DFA with output as follows: Let the state space be $Q=\{q_i \mid 0 \le i \le 9\}$, and input and output alphabet be $\Sigma = \{i \mid 0 \le i \le 9\}$. The initial state would be $q_0$. Let the DFA read the decimal number in reverse. For any state $q_i$, on reading $d$, you move to state $q_j \in Q$ and output $k \in \Sigma$ if $10j + k = 2d + i$ (Why can you always find such $j,k$?).
Basically, you are trying to store the carry while outputting the least significant digit of the multiplication of the current digit by 2 after adding the last carry, just as the grade-school multiplication.
Then, you can readily create a TM using this DFA with output that does the required multiplication.