I have been an on and off player of the FTP level and have gone as high as i can on 2 different SF chracters. However, many of the missions result in overwhelming domination by the AI in combat regardless of my ships' layouts (weapons, shields, maneuverability). I've heard backtalk that it play much different as a lifer.
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The veteran ships are quite nice and would work wonderfully for someone who hasn't already purchased a good C-store ship or two (or three, or...). Heck, someone could easily use the veteran ship and never buy a C-store ship at all.
But, as for the basic question, I think the only game play advantage a lifer might have is that having spent that amount of money on the game, a lifer is possibly more likely to take some extra time to hone their skills/set ups compared to a casual F2P. We definitely do not get any kind of damage or resistance bonus for being a lifer and, with the exception of the captains yacht already mentioned, what little gear and stuff a lifer gets, (that might have been top notch when LTS first came out), isn't even good enough to be called second rate compared to current C-store and even free stuff in the game.
The LTS has no affect on gameplay. Sure, Lifetimers get some perks like T5 and T6 veteran ships, +5% extra XP, and an android bridge officer, but they have no real affect on gameplay. I can play STO just find with either my primary STO account (purchased LTS) or my free 2 play only secondary STO account.
The problem you are having is likely based on the gear you have, the bridge officer (Boff) abilities you have slotted, and your own skills as a player (everyone goes through this phase).
The likely issue is that you are not used to having your shields drained to nothing in a matter of seconds. The best defense against the Borg is "hull tanking" which means to focus your attention on hull repair while in combat. Don't bother using abilities to restore your shields when the Borg's tachyon beam which is used to drain shields is still in effect. You want to basically focus on Engineering Team and Hazard Emitters. You can also Polarized Hull since it provides some damage resistance.
You should check the Exchange for a Mk X+ Neutronium Alloy if you do not have that engineering console yet. It gives you resistance to all energy weapon and kinetic damage. Get the highest Mk your level allows you to use. Even a common (white) version is better than nothing.
Beyond that try and stick to one type of energy type as the damage boosting consoles wok better the more specialised they are, Ie +Phaser consoles will do a lot higher percentage (+50%) than +beam damage consoles so help damage output. If your enemy is dead it cannot shoot you is a mantra for pretty much most players here today. Also the Defiant type ships, ie escorts, are squishy so you may like to swap to a cruiser first, then maybe a science vessel before moving on to escorts.
Still waiting to be able to use forum titles
as mentioned, it's also pretty important to kill stuff as fast as you can. with that in mind, the specialized damage type tactical consoles work better than the more generalized beam or cannon ones. so, for example, if you're using a full rack of phasers and phaser damage consoles, you'll do that better than someone using a mix of different damage types and beam consoles.
further, like dirlettia said, different types of ships are different. cruisers will be much more durable than a comparable-tier escort or science ship, though they will have less potential damage or potential support capability in the sciship's case. if you're getting rekt in an escort, try fitting up a cruiser comparably and doing the same content. keep in mind that with a cruiser, you can equip more weapons than on an escort, and because they're less maneuverable, you'll probably want to circle your prey, keeping them in a broadside arc so that all 8 of your beams can fire on them at the same time, rather than making strafing runs like you would in an escort.
my final advice here: if you don't have a fleet, get one. if you have a fleet but noone's ever on, join an active one. even if you prefer to play by yourself and not talk much, there are very solid reasons to join one. not the least of which are access to fleet stores and the ability to buy fleet ships. furthermore, there are a lot of sto veterans around willing to freely share advice and tips and other assistance where it's needed. if you're interested, the 44th Fleet is always looking for new friends. pop on over to startrek.44thfleet.com and toss an application up on our recruitment forum.
good hunting o7