Python 2, 129 126 119 bytes

def f(n,k,p=0):z=p<1or n>0;q=int('7367355777e0d93bf0fb'[n%10::10],16)*z;return(z and f(n/10,k,q))+k*bin(p^q).count('1')

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As specified, takes a non-negative number and a positive sign length, and returns total changes.

Python 2, 129 126 119 104 bytes

def f(n,k,p=0):z=p<1or n>0;q=-~ord('}/lx2Z^o~z'[n%10])*z;return(z and f(n/10,k,q))+k*bin(p^q).count('1')

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Thx for a big 15 bytes from ovs.

As specified, takes a non-negative number and a positive sign length, and returns total changes.

Python 2, 129 126 119 bytes

def f(n,k,p=0):z=p<1or n>0;q=int('7367355777e0d93bf0fb'[n%10::10],16)*z;return(z and f(n/10,k,q))+k*bin(p^q).count('1')

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Python 2, 129 126 119 bytes

def f(n,k,p=0):z=p<1or n>0;q=int('7367355777e0d93bf0fb'[n%10::10],16)*z;return(z and f(n/10,k,q))+k*bin(p^q).count('1')

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As specified, takes a non-negative number and a positive sign length, and returns total changes.

Python 2, 129 126 111 bytes

def f(n,k,p=0):q=int('7367355777e0d93bf0fb'[n%10::10],16)*(n>0);return(n and f(n/10,k,q))+k*bin(p^q).count('1')

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Python 2, 129 126 119 bytes

def f(n,k,p=0):z=p<1or n>0;q=int('7367355777e0d93bf0fb'[n%10::10],16)*z;return(z and f(n/10,k,q))+k*bin(p^q).count('1')

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