Here are two schedules that should work. They are symmetric, so rows & columns can be assigned to rounds & games either way around.
14 teams (A to N), 7 rounds/7 games
(K G) (B N) (J D) (C H) (A L) (F E) (I M)
(F N) (I D) (C L) (A G) (K E) (B M) (J H)
(J E) (A M) (F H) (B L) (I G) (C N) (K D)
(I L) (C E) (K M) (F D) (B H) (J G) (A N)
(C M) (K H) (B G) (I E) (J N) (A D) (F L)
(A H) (F G) (I N) (J M) (C D) (K L) (B E)
(B D) (J L) (A E) (K N) (F M) (I H) (C G)
16 teams (A to P), 8 rounds/8 games
(A P) (C O) (L B) (H K) (I J) (M N) (F G) (E D)
(H D) (L G) (C J) (A N) (F B) (E K) (I O) (M P)
(M J) (I N) (F D) (E G) (C P) (A O) (L K) (H B)
(F N) (E J) (M G) (I D) (H O) (L P) (A B) (C K)
(C G) (A D) (H N) (L J) (M K) (I B) (E P) (F O)
(E B) (F K) (I P) (M O) (L D) (H G) (C N) (A J)
(L O) (H P) (A K) (C B) (E N) (F J) (M D) (I G)
(I K) (M B) (E O) (F P) (A G) (C D) (H J) (L N)