Heavy Escort Carrier is the generic name for the Akira/Armitage-class variants. Look for Armitage or Heavy Escort Carrier in the C-Store. It's a slightly Engineering-heavy escort with a single hangar bay.
There's the Atrox, which is a big lumbering hulk, somewhat like the Vo'Quv. And the Heavy Escort Carrier, which is pretty much what it says on the tin: a slightly bulky escort with a carrier bay.
Apart from those, there's the lockbox ships: Jem Hadar Escort Carrier and Dreadnought, the Recluse and ehm, yeah. I feel like I'm forgetting one, but I'm not sure.
And indeed: engineers are something lethal in ground combat. It just takes a whole lot of work to get them up to par in space (though there are people around who manage it, mind you). *shrug* I just started a second KDF engineer, mostly because I already had 2 sci's and tac's but only 1 eng, but also to see how he would do when specced to BoP style of play, with the option to go NeghVar if that didn't work out. The energy management options an engineer gets are nice: my fed Galaxy Dreadnought captain tends to run close to max power in all subsystems, and that's without running an aux2bat build. You just need to figure out what to do with all that juice, if raw cannon output isn't it.
One more note about the HEC. If you're sticking with the Engineer character, you're already Eng heavy. If you go with the HEC, you'll probably want to use that Lt. Cdr Engineer slot for something damage dealing like Eject Warp Plasma or preferably Directed Energy Modulation. You really don't need the extra tank, what you need is damage. Running Elite Scorpion Fighters (from New Romulus reputation) in your hangar bay would help add a bit of DPS as well.
You can buy it from the exchange as well, and that is arguably the safest way to get it.
The downside is its pricetag: last I checked it was some 60 million EC.
On the other hand, when you buy it from the lobi store (which is where it is found, not directly in a lockbox), it costs 800 lobi (I think. Might be 600). Lobi you can get only from opening lockboxes, and every lockbox contains at least 4 lobi (average over hundreds of boxes was around 5 per box). So ehm, yeah. If you're going that route, you might want to consider buying Master Keys with Zen and selling those for 1,4-1,5 million each, as that would likely be cheaper.
Edit: keep in mind that the lockbox ships are a 1-character only unlock. If you want them on another character too you have to pay for that one too. I tend to prefer c-store ships for that reason (either with fleet upgrade or not)
My Eng hit level 50 recently last week - she was, for lack of a better term, on the reserve bench; I only really logged in as that character when wanted to replay story content.
Though now she is at level 50 I find that I just can't face restarting the grind from the onset. My (primary) character, a sci, has T5 rep on all but Nukara (currently 3/4 way through T2 of that), but I hit a brick wall with my second character (a tac) and just can't face starting from scratch with my eng at all.