K (ngn/k), 41 32 bytes

{,/(|4,&0|,/-:\)'-':+(0,y)\x-97} 

Try it online!

output: 0=D 1=R 2=U 3=L 4=!

{ } function with arguments x and y

x-97 convert "a".."z" to 0..25

(0,y)\ divmod by y, returns a pair of lists - one with the quotients and one with the remainders

+ flip - make it a list of pairs

-': deltas - subtract each pair from the previous one; use 0 0 as an implicit pair before the first

( )' for each pair (let's call it (Δi;Δj)) do:

• -:\ self and own negation: (Δi;Δj) -> ((Δi;Δj);(-Δi;-Δj))

• ,/ concatenate: -> (Δi;Δj;-Δi;-Δj)

• 0| max with 0

• & "where" - for instance it would turn the list a:1 5 0 2 into 0 1 1 1 1 1 3 3, i.e. each index i is repeated a[i] times

• 4, prepend a 4

• | reverse

,/ concatenate