Air carrier Captain reported a ground traffic conflict with an aircraft blocking a taxiway preventing exiting a Runway while another aircraft was on final approach. The flight was finally able to contact ATC and received clearance to taxi and exited the Runway safely.

Narrative

Probable Two Aircraft on XXR at the same time: The Tower was very busy; even though it was the middle of the night...the MOST busy I have seen at ZZZ. There was no Ground Control; (or probably Clearance Delivery); so Tower had to do everything. Tower cleared us to land on XXR. Landing was uneventful. Tower cleared us to taxi 1; Cross Runway XXL; to [taxiway] 2; to parking at 4; pass behind Aircraft Y (who would be taxiing in front of us). I turned left on 1. Just after I turned; Aircraft Y stopped right in front of us; in the middle of taxiway 1; on 2. Later the FOs told me they shut off their lights. I saw them stop in front of me; so I stopped just short of theXXL hold short line; as I didn't want to be stuck in the middle of the Runway. Then I looked back; and saw an aircraft on very short final for Probable Two Aircraft on XXR at the same time: The Tower was very busy; even though it was the middle of the night...the MOST busy I have seen at ZZZ. There was no Ground Control; (or probably Clearance Delivery); so Tower had to do everything. Tower cleared us to land on XXR. Landing was uneventful. Tower cleared us to taxi 1; Cross Runway XXL; to [taxiway] 2; to parking at 4; pass behind Aircraft Y (who would be taxiing in front of us). I turned left on 1. Just after I turned; Aircraft Y stopped right in front of us; in the middle of taxiway 1; on 2. Later the FOs told me they shut off their lights. I saw them stop in front of me; so I stopped just short of the XXL hold short line; as I didn't want to be stuck in the middle of the Runway. Then I looked back; and saw an aircraft on very short final for XXR; with another behind. I told my PM; that we needed to get ahold of Tower and tell them we were not clear of XXR; because there is not enough room between XXR and XXL for a larger airplane to stop; and be clear of both Runways. Tower was very busy; and the PM could not get a call in immediately. I looked back; and the closest aircraft was now flaring and landing on XXR. I was seconds away from taxiing on to XXL; enough to clear XXR; and then figure out what to do. I didn't want to cross and taxi on R; without a clearance; and because I didn't know what Aircraft Y was going to do; and I didn't want to collide with them on 2. Just after that; my PM got ahold of Tower; and told them what happened. Tower cleared us to cross and taxi on 5; so I accelerated relatively quickly to clear XXR; and then continued normally across XXR; and up taxiway 5; Right on 3; to parking at 4; uneventfully.Cause: The key problem was that Aircraft Y; who was cleared to taxi on 2 (and I assume hold short of XY on 2); stopped on 2; directly ahead of us; blocking taxiway 1. We could not follow our ATC clearance; which was to cross Runway XXL on 1; and turn Left on 5. Contributing was the fact that it was nighttime; and Tower was VERY busy. So; Tower had; practically; no chance to see that Aircraft Y stopped on 2; at 1; blocking our taxiway; and the landing aircraft on short final; would have had a hard time seeing that we were not all the way clear of XXR; as they were landing.Suggestions: For me; it would have been better to taxi onto XXL immediately. I could have stopped between taxiways 6 and 1; on Runway XXL; and waited; clear of XXR. If it was daytime; and/or Tower was not so busy; they may have been able to see Aircraft Y stop on 2; blocking 1; and given us a new clearance; and/or told the aircraft on short final to go around; or both. The Landing aircraft; could have seen us on Runway XXR; and gone around. This would have been MUCH easier to do in daytime. When I am PM; and there are aircraft going to be ahead of us on the landing Runway; I tell them I will give them a 'Runway Clear' call; as soon as the Runway is clear and safe to land. If I am PF; in the same scenario; I tell my PM to give me a 'Runway Clear' call as soon as they see the Runway is clear and safe to land.