Generally, if you find yourself taking heavy casualties to a numerically inferior fleet, the enemy probably has an Area of Effect (AoE) damage ability you are not countering. The Marza's missile barrage is a classic example; if you do not stop it with an ion bolt or detonate antimatter, it can deal over 3000 damage to every ship in your fleet, I.e. killing all your frigates. Take the time to learn which ships to watch out for so you can counter them appropriately.

So did you manage to win with the Advent? One thing about the TEC is that while the TEC economy becomes amazing late game, the TEC units overall do not get as powerful as the Advent and Vasari ones late game. The TEC loyalists are really only geared to win a battle of attrition, where they can use their superior economics to fund double starbase defenses to wear down the enemy while striking them with Novalith cannons, which will hurt the enemy economy hard. All else being equal on the offensive and maybe even while defending you will lose to Advent and Vasari late game fleets, so you need to get used to using your massive economy to replace your losses. They basically fight like Russians with America's trade and industrial power.

Since you're playing E4X, make sure you get ship upgrade tech so you can give your Ankylon the shield draining beams. The constant shield healing will help it a lot and will improve its survivieability even if it runs out of antimatter.

What level AIs are you playing with? On the higher difficulty levels the AI does get economic bonuses, so it may well be building fleets faster than you can. Still, since you are a human with superior management skills (hopefully), it is more than possible to beat even unfair AI in a war of production. On the higher (purple) difficulties, it is almost impossible to out play them normally, you have to exploit units the AI does not handle very well like starbases and tons of bomber hit and run attacks to inflict as much free casualties as possible to even the odds, and slowly wear them out.

Destra's are pretty good heavy cruisers. Basically they are a jack of all trades ship that is good but not great against all the other combat frigates in the game, and are extremely durable to everything except bombers. However, other combat frigates are more efficient for raw DPS, though the Advent's plasma upgrades make the Destra a bit more competitive. If your opponent doesn't have a lot of bombers, they are a good general purpose ship to add to your fleet when you don't need to counter anything specific.

Defense vessel are "flak" frigates, and are a key part of the main combat "square" of Light Frigates > Flak Frigates > Corvettes > Long Range Frigates > Light Frigates etc. Basically build them when you're enemy has a lot of corvettes or strikecraft (especially fighters). They're also extremely durable for how cheap they are, but they struggle to damage anything larger than a Long Range Frigate.

Dominas are a support cruiser, and they are much better in E4X than they are in Vanilla, so some things you may read about them may not apply. They are used for both healing your ships and for disabling enemy frigates. As the Advent Rebels (Ascended Advent) in the mod, you do not have to research these ships, which gives them some interesting uses. For example, they are a huge help when taking Terran, Desert etc. planets that have a few heavy cruisers, since they can just disable the heavy cruisers while the rest of your fleet kills the weaker ships, then fight them alone. I typically build one for each capitalship/titan I have, since the healing ability you research both helps you keep them alive and prevents abilities like Ion Bolt from disabling your key ships.

The Vasari have an inherit advantage against the advent thanks to their phase missiles, which bypass shields, one of the Advent's great technological strong points. Some research, pacts and hero units can help with this weakness a little bit, but its definitely an uphill fight.

The good thing about the Vasari starbases is that they will move out of range of repair bays and other defensive structures, allowing you to engage them alone. Of course taking out an enemy fleet WITH a starbase is a much more difficult proposition. You either need to provoke the AI into attacking you (retreat your fleet from the front line, wait for them to attack your now vulnerable planet, warp right back in) so you can destroy the fleet, or simply greatly outgun the enemy.

If you're struggling on normal AI, you're probably making to fairly big beginner mistakes. First, you're probably not expanding fast enough early game. Taking planets from the militia is much easier than from other players, so early on invest only in the basic research infrastructure and build up your fleet to take as many easy planets as possible (minimum you should max out your starting fleet supply quite quickly). Strongly tied to this is the strategy of double colonization - that is, building a colony capitalship to expand in one direction, while frigates and a colony frigates expand in another.

The other mistake many beginning players make is either not building a big enough fleet or building two many heavy cruisers. A few dozen frigates is not a big fleet in Sins. While one must be careful to not expand their fleet size too quickly do to the permanent economic cost, at the same time when you max out your existing fleet and you're still struggling, never be afraid to expand it. A good economy is no good to you if you let your enemy destroy it.

Alternatively, a fleet ratio of 50% capitalships to 50% frigates is only acceptable at the beginning of the game. For their fleet supply, capitalships must be very high level to match the efficiency of frigates in basic stats. The main reason you should build them is for their special abilities, not to increase your fleets damage output etc. Further, the more capitalships you have, the slower they and your titan will level up, since all experience earned is split. While it doesn't sound like you're doing this, if you have multiple Radiance's etc. per fleet, just keep in mind that capitalship quality is usually better than quantity. Level 10 capitalships do not suffer from any of these issues, so feel free to build more once some of them start hitting lvl 10 as well.

Culture is more powerful in E4X, especially for the Advent Loyalists (Orthodox). Occasionally I'll see the AI build 5 or more culture centers on one world, in this case the usual 1 culture center for every few planets won't be good enough. You either have to destroy them quickly, build more culture centers yourself, or both. Also capitalships and titans can repel enemy culture in friendly territory, so if you have some to spare station them on the culture front lines.

Well, playing as TEC Loyalists without superweapons is going to be an immensely harder task. So much of their strategy relies on it that I would recommend playing the TEC Rebels (HTS in the mod) instead.