Well, for starters, I'd say don't use D&D. It is a game tailored towards violent conflicts, which is exactly what you're avoiding, it seems. Mind you, I said "violent conflicts". No story, thus no game, can exist without any conflict whatsoever. I'm not also saying it's completely undoable with D&D, just mainly... a waste of its design and practical goals. Another way to put it, to use a metaphor, is: smartphones are great, you can do a lot with them, they're like handheld computers... But they can't really substitute a desktop computer in every way, maybe not even most ways.
Now, if you're willing to work outside of D&D, there are some good systems out there for that "action and adventure doesn't mean swinging swords all the time" vibe you're after, like, for example, Fate (The Dresden Files RPG, Spirit of the Century, Diaspora, etc), in which most of the mechanics about resolving conflicts are the same, regardless if it's a brawl, a wardrobe and style show off or even an economic dispute between Lex Luthor and Bruce Wayne. It's worth a look, really, and there are SRDs available for some of those games (Spirit of the Century and Diaspora, iirc).
If you're sticking to D&D, plotting the campaign isn't the difficult part; the difficult part is to design encounters (which is just a way of saying "conflict scene") that allow your PCs to shine doing their thing when most of their sheets are geared towards combat (yeah, players do that, it being the game it is).
For the wizard (most hocus-pocus folks, really) and rogue, that's easy. Most other types, though, will rely purely on RP, most of the time, which isn't bad per se, just kinda unfair, since some players get to look at their sheets and say "I can do this, this and that", while the fighter's player has to memorize lines from Gladiator and the ranger's player has to become a living Bear Grylls encyclopedia.
If your group can pull it off, that campaign'd be the stuff of legends, but it'll be hard, really.
I think there are some suggestions I can offer, but first I wanted to check with one of the comments that you have made.
You say that characters with a d6 in shooting can't hit. This seems a little off because the odds for hitting a TN of 4, i.e. a target at short range with no cover, are 75% for a Wild Card. I certainly have players in my games with d4s and d6s in shooting that have no problems with hitting, especially once you take into consideration how frequently a d4 will explode (1 in 4 chance).
Assuming that you are applying the rules properly, I can suggest a few things that will improve the situation:
Melee Combat
Ganging Up - This gives +1 to hit for every ally in combat with your foe up to +4. Even if a character isn't hitting they can assist with giving bonuses to other people.
Push Manoeuvre - This relies on the Strength Attribute rather than Fighting Skill, and allows you to move the target around, cause damage if you have a shield, or knock the target prone.
Wild Attacks - These give +2 to hit and +2 damage, and are extremely effective. Watch out for the reduced Parry you get though.
Smarts/Agility Tricks - These have a similar effect to Taunt and Intimidate, but are more dependent on good teamwork as the -2 Parry on a simple success only lasts until the affected creature's next turn.
I'd try and be as flexible as possible as GM when players come up with ideas for tricks. One of the best examples I've ever had is with someone who grabbed a can of baked beans and threw it whilst shouting 'grenade!' - as a successful smarts trick it had all of the enemy diving for cover
Ranged Combat
Aiming - Do nothing for a round whilst you aim and you get +2 to hit, which is very effective.
Using a shotgun (depending on the setting) - gives +2 to all shooting rolls if you fire a single barrel. Makes you more likely to hit an innocent bystander though.
You've also got double tap (+1 to hit and damage), and three round burst (+2 to hit and damage) with certain weapons.
Other general comments
There are a few general Edges that can assist with low skill combat characters. One that springs to mind is Elan, which gives +2 to any roll you make as a result of spending a Bennie.
Consider giving non-combat characters things to do whilst others are fighting. Dramatic tasks are good for this, but get a bit old if overused. For example, your computer hacker is trying to break into a security system so you can escape the room you're in whilst the rest of the group are holding back the drug enhanced super mutants.
Also don't forget that a successful trick, taunt or intimidate with a raise will shake your opponent, making them MUCH easier to wound. For high toughness creatures this becomes a hugely valuable tactic.
Also, there are a couple of general resources that you might find useful:
Savage Worlds Combat Survival Guide - which summarises the options you have, along with suggestions as to when each of these might be used. It should be noted though that there are a couple of new options introduced in the Deluxe version of the rules, e.g. pushes, that are not covered on this sheet.
Whispers From the Pit - Expanded Trickery - which goes into details about tricks and how to run them as GM. ... looks like this site has been corrupted
Best Answer
As a fan of the Suikoden franchise, I also like creating basic NPCs that follows the players (the Smith is a common one).
You seem to already avoid the pitfalls of most GMPCs: disliked by PCs, and abnormally strong and awesome. That's great.
Since your problem is mostly related to combat, I would suggest an option I use when NPCs become good enough to be useful in combat (otherwise they stay in the back, or simply give a passive bonus): let your players play them.
Make a character sheet, control the character out of combat, and when combat happens simply hand the sheet to the players. Either have one player control it (the one most familiar with the rules, or whose character cannot play at the moment), or even make it a group thing with all players (you included) deciding what the NPC should do.
For lazy players, consider statting the NPC as a monster, with a couple at-wills and some recharge powers, just like a monster. The (significantly) reduced complexity of the character might make them less reluctant to take on the additional effort of managing the NPC's combat.
However it's statted, handing over control of the character makes it easier for you to avoid most bias, and for the players to better relate to the NPC.