Advice vs. Meso Civs

I'm a pretty new player, ~1300 DE ELO after a few months. Looking for some advice to help reorient the way I'm thinking about match-ups vs. Meso civs, particularly Aztecs and Mayans.

In almost every game I play, I either win or it's obvious to me why I lost -- didn't understand my opponent's strategy, too slow to eco up, wrong choice of unit composition. These are easy to learn from and fix. But against Aztecs and Mayans it feels like I play well and then get totally rolled every time, which is especially weird since the two of them ostensibly are so different, and yet it all feels the same to me. For this reason I suspect there's something very wrong in the way I'm thinking about playing against these civs.

Note that I exclusively play Persians (for now, picked one versatile civ to focus on while I'm learning). I realize that this immediately changes the conversation, since it's the only civ without access to Two-Handed Swordsman, but I assume the answer can't just be "yeah Persians should always lose to Aztecs/Mayans".

I could post replays, but I don't want to get bogged down in the individual games. What I'm looking for is generalizations. What's a reasonable game-plan? Here are some specific questions to illustrate my confusion. Feel free to comment on any, all, or none of them.

• Against Mayans, of course I expect archers. However, their archers are so strong that if I just try to defend with skirms or knights I always get overwhelmed. Recently I've been mixing in onagers, which tends to push them back -- until they switch into eagles a few minutes later, and then I'm helpless.

• Against Aztecs, I know they're supposed to have a weak post-Imp, so I usually try to just hold off until then. However, Aztec players invariably put on heavy Castle Age pressure. I can never seem to get the balance right between building enough of an army to defend successfully in Castle, and booming enough to not find myself at an irretrievable disadvantage in early Imp. I never have this problem against other civs that often go for castle age pressure like Mongols or Spanish...

• Neither civ ever feels vulnerable to me. They're both above-average in Feudal Age. In Castle Age, Mayans either go for obsidian arrows crossbows (which usually has to be dealt with with an equally large number of skirms; easy to get caught off-guard once with too small an army and then the game is over even if you walled) or more often, plumes, which never have to take a bad fight except against cavalry, and in numbers are too strong against knights (even with Persian knight bonus vs. archers) for that to be relevant. Aztecs in Castle Age have eagle warriors, which are countered by only exactly knights (again, playing Persians; going Militia-line not a real option), which is not great against the civ with the best monks in the game.

• In spite of these strengths, attempts to out-boom these civs invariably meet with disaster. Presumably my opponents know I can't touch them and just go crazy on eco while I'm forced to build military in case the go on the offensive, but can't go offensive with it myself. By early Imp, the game is over. I know hand cannon is supposed to be a strong counter to Eagles, but it's too late by then.

I expect to be called an idiot -- hopefully I really am just overlooking something or thinking about something very wrong. Please, if you know what it is, tell me!

Thanks.