I've started a new character, and I'm considering having her be a marksman who also fights with a short sword in close combat. She doesn't use a shield, as she finds them too clunky.

I was thinking of taking the obvious Warrior birthsign, and customizing the class to include blade, security, sneak, light armor, and marksman.

Is this a good build?

Certainly. You might also consider the Thief birthsign, for your Sneak, Light Armor, and Marksman (as well as for the Luck.)

Thief sounds good for me. I've always been prone to select the Warrior sign, but this time I want to try something different. Light armor sounds good.

I'll also cheat a little and use the duplication glitch so she'll have countless arrows at her disposal.

I'm going to offer different advice: focus either on an archer or a warrior, but not both. Not unless you intend to game the leveling system.

I currently play as an Archer who can toss a few destruction spells, but it's been difficult to level both so that, in close battles, I can burn or freeze my way out of a tough situation. My Marksman now sits in the 70s, while my Destruction's in the 40s. Restoration, because I heal myself quite a bit, is in the 60s.

If you do try to level out both, and increase levels, you're going to be at a disadvantage in fights, especially when you're outnumbered. It'll be a difficult situation, but if you've the cunning to do so, you could pull it off. I just find those caves to be too narrow to have a fight in which I can use both attacks effectively, relying more on my distance for protection than a fire spell which is only strong enough to pop corn.

I would suggest if you go this route, clear Cyrodiil of mud crabs with your blade and the rest with your pinpoint arrow shots through the heart.

As noted - with the vanilla leveling and a straightforward "most important skills as majors" build, marksman/melee is a tough thing to play. The problem is that the marksman skills are all stealth and the melee skills are all combat, and you can't pick both as a specialization. Whichever one you pick - those skills are going to increase generally quicker than the others, and like it or not, your character is going to start drifting in that direction.

If you really want to combine both, the most effective thing to do is to use combat spec, then make your majors marksman, light armor, sneak, acrobatics and some assorted filler stuff - mercantile and restoration are always good choices, and I like Mysticism or Destruction. Do NOT pick your melee skill as a major since, with a combat specialization, it'll increase far too quickly and your other skills won't be able to keep up, and you'll end up with just another melee fighter. The downside though is that that means that your blade skill will start out fairly low, so you'll have to either train it on your own or pay for some training to get it started. Once you get it moving though, you should be able to just go through the world and do whatever you do and everything will take care of itself from there on out.

The whole key - not just to this build, but to any really effective one (given Oblivion's warped leveling system) - is first to understand that specialization is the single most important character design decision, and then to pick the one that works best to allow you to split your important skills up into some combination of NON-specialized majors and specialized MINORS. That's crucial. When you're done, you want to be able to look at your skill list and pick out the most important skills for your character and see that as many of them as possible are either specialization MINORS or NON-specialization majors. If you do that, then you can generally just go and do whatever you do and the rest will work itself out.

Hmm...good thinking. I honestly never thought that specs and major/minor skills were that critical in forming a good character that wasn't either too powerful or too weak.

For an archer, it'd be better, like you said, for blade to not be a major skill. Instead, maybe I should do marksman, security, sneak, acrobatics, mercantile, destruction and restoration? But keep the spec as combat.

EDIT: OK, so it would seem I had forgotten to save at the sewer entrance after making the character. I suppose I could still pull it off, right? Or do I need to start over?

With the build you listed in the opening post? What's the specialization?

Seriously - specialization is THE most important decision in the entire process. Specialization skills increase up to twice as fast as non-specialization skills, which makes for a hefty difference.

With Stealth spec, it should be fine. Security and Sneak will be the only potential problems - Sneak because it increases a bit too quickly and Security because it's about impossible to not use it (unless you use Open spells) and increasing it doesn't do anything at all to help you stay alive when tougher opponents start showing up. Light Armor tends to lag a bit for archers, since they spend a certain amount of time outside of melee reach, and thus not getting hit, so it should be okay as a spec major (and I've sometimes played it that way for just that reason). And Marksman is one of the slowest increasing skills in the game - even as a spec major, it's generally fine.

With Combat spec, Blade could be a problem, but everything else will be okay. It just means that you'll have to focus on Marksman more, since Blade as a spec major will increase much faster than Marksman as a non-spec major. If she spends most of her time shooting and only sometimes falls back to melee, it should be fine. If she jumps into melee quickly, Marksman is going to start to lag.

I wouldn't worry about it too much in either case. It's not a bad build either way. If it really bothers you, then without a sewer exit save, you'll have to start over, but if it was me, I'd just keep going and see what happens, and if the character starts to skew one way or another, that just becomes part of her roleplay.