R, 22 19 bytes

Producing two different sequences with the correct limiting frequencies whether \$s=0\$ or \$s\ne 0\$:

(1:n%/%10^(!s))%%10

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As for checking whether or not the merged sequences work as well, one can try

summary(as.factor((10*((1:n)%/%10%%10)+((1:n)%%10))))

R, 22 19 17 bytes

Following earlier resolutions, this R code is producing two different sequences with the correct limiting frequencies whether \$s=0\$ (penultimate digits) or \$s\ne 0\$  (last digits):

1:n%/%10^(!s)%%10

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As for checking whether or not the merged sequences work as well, one can run, e.g.,

summary(as.factor((10*(1:n%/%10%%10)+(1:n%%10))))

R, 22 bytes

Producing two different sequences with the correct limiting frequencies whether \$s=0\$ or \$s\ne 0\$:

((1:n)%/%(1+9*!s))%%10

Try it online!

As for checking whether or not the merged sequences work as well, one can try

summary(as.factor((10*((1:n)%/%10%%10)+((1:n)%%10))))

R, 22 19 bytes

Producing two different sequences with the correct limiting frequencies whether \$s=0\$ or \$s\ne 0\$:

(1:n%/%10^(!s))%%10

Try it online!

As for checking whether or not the merged sequences work as well, one can try

summary(as.factor((10*((1:n)%/%10%%10)+((1:n)%%10))))

R, 21 bytes

Producing two different sequences with the correct limiting frequencies whether \$s=0\$ or \$s\ne 0\$:

(((-3)^!s)*(1:n))%%10

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R, 22 bytes

Producing two different sequences with the correct limiting frequencies whether \$s=0\$ or \$s\ne 0\$:

((1:n)%/%(1+9*!s))%%10

Try it online!

As for checking whether or not the merged sequences work as well, one can try

summary(as.factor((10*((1:n)%/%10%%10)+((1:n)%%10))))