J S R Y O D
E L A M N F

Of your 2nd die, I don't see any way to avoid using each of F,L,N exactly once, thus violating rule #5.

JA = Jan
FY = Feb
MR =Mar
AR =Apr
MY =May
JE =Jun
JL =Jul
AS =Aug
SM =Sep
OR =Oct
NO =Nov
DE =Dec

JRYSND
AMEOFL

Jx3 Rx3 Yx2 Sx2 Nx1 Dx1
Ax3 Mx3 Ex2 Ox2 Fx1 Lx1

First letters month.
J 3x
A 2x
M 2x
F 1x
S 1x
O 1x
N 1x
D 1x

Fixed months:
JE = june
JL = july
MY = may

One die will have J, then the other will have e,l

J * * * * *
E L * * * *

Each die need to be first letter 6 times. Die 2 has E and L which are never first letter of the month. So to make 6 times first letter with the remaining 4 letters, we need both A and M twice, plus 2 letters of FSOND.
As M is on die 2 Y has to be on die 1.

J Y * * * *
E L A M * *

April has no second letter that can be a first letter. So die 1 has one of p/r/i/l and 3 of FSOND.
August needs a second letter on die 1. This can only be S.
March needs a second letter on die 1. This can only be R.

J Y S R * *
E L A M * *

January has to be Ja because it can only use a letter from ELAMFOND. Only A will give a unique set.

Months covered so far:
January = Ja
March = Mr
April = Ar
May = My
June = Je
July = Jl
August = As

On die 2 L can't be in any months except July: L = 1.

One of FOND has to be used twice on die 2. Only O can be used twice: October(O.) and nOvember(No). So O goes on die 2 and N on die 1.

J Y S R N *
E L A M O *

Both F and D can only be used once on either die.

Letters used so far.
1 2
J a (january)
J e (june)
J l (july)
y M (may)
s A (august)
S . (september)
r A (april)
r M (march)
N o (november)
. O (october)

Die 1: J(3), S(2), R(2), Y(1), N(1), F/D(1)
Die 2: A(3), M(2), O(2), E(1), L(1), F/D(1)

On die 1 Y or N needs to be used again. Only option is Y in february. So F goes on die 2 and D on die 1.

Die 1: J Y S R N D
Die 2: E L A M O F

Just for completeness.

Letters used.
1 2
J a (january)
J e (june)
J l (july)
y M (may)
s A (august)
S . (september)
r A (april)
r M (march)
N o (november)
. O (october)
y F (february)
D . (december)

Die 1: J(3), S(2), R(2), Y(2), N(1), D(1)
Die 2: A(3), M(2), O(2), E(1), L(1), F(1)

October needs to use one of RSY on die 1, to make it 3. Only option is R.
September/December can use E and M to get those to 3 and 2. So they either use one E and one M or both use an E.

[edit] For rule 6 *cough*: O and A have both been used as first and second letter, so the others can't be used as both first and second letter. September/December need a second letter on Die 2. This can only be E as M has been used as first letter.

Answer:

Die 1: J Y S R N D
Die 2: E L A M O F

Die 1: J(3), R(3), S(2), Y(2), N(1), D(1)
Die 2: A(3), M(2/3), E(2/3), O(2), L(1), F(1)
[edit]: Die 2: A(3), E(3), M(2), O(2), L(1), F(1)

JA FY MR AR MY JE
JL AS SE OR NO DE

JRSYDN
AEMOFL

"Y m (may)" (twice) should be "y M (may)".