Late March: There will be more domestic and foreign tourists than in mid-winter, but it is still not a busy time for leisure travel so wherever you choose should be manageable. I would still tend to avoid weekends at Badaling, just to be sure. But Badaling at this time of year during the week, especially if you just walk on the way about 20 minutes away from the main access point, should be acceptably free of huge crowds. There are definitely ways to manage and get a good GW experience at Badaling, if that turns out to be your most cost-effective option.

It will still likely be chilly but more jacket or windbreaker weather rather than coat weather. Still, with kids 10 and 11, I would not plan on long day hiking but maybe a couple of hours along the Wall itself is enough.

Mutianyu: The direct bus from Dongzhimen bus station in Beijing to Mutianyu Great Wall unfortunately will likely not be running, as it usually starts up again around mid-April. Mutianyu by public bus will be cheaper on your wallet than a private car, you just have to be willing to spend a couple of extra hours-ish overall on the excursion, due to lower travel efficiencies. Metrics based on 2011 prices (for simplicity sake, I will treat each child as a full adult fare, you might get away with a cut rate if they are small, China does child fares by size not age):
--Public bus Dongzhimen to Huairou: RMB 17 x 4 = RMB 68.
--Minibus Huairou to Mutianyu GW: RMB 20 x 4 = RMB 80. You might be able to find an empty that will leave immediately for your family, without waiting for others to fill the minibus, for about RMB 100.
--Minibus back to Huiarou: RMB 80 total
--Public bus Huairou-Beijing: RMB 68. Total of around RMB 300 for transportation. Expect to spend about 2.5 hours each way for this transport sequence, vs 1.5 hours by private car costing RMB 600-700. So value proposition: is RMB 300-400 savings worth 2 to 3 hours of your time? Only you know your Beijing schedule and can answer that. Entry fees are extra of course, and so are cable car fees if you use. This will go for any other GW location as well.

Badaling: By far the cheapest DIY excursion, as train tickets from the conveniently-located Beijing North (Xizhimen) Station are less than RMB 20 per person for the best seats each way, about half of that for the standard seats. So you could manage total transport for the whole family to/from Badaling for in the range of RMB 90 to 140. And you can buy those tickets daily, on a walk-up basis. Though train frequency is cut back in the winter, by late March they should have the usual decent selection of times back--you can doublecheck when you get to Beijing. I recommend you seriously consider this option if you are on a tight budget.

Do take a look at those hostel tours also though. They will provide full round-trip transport to/from Beijing, and also usually the entry fees, guide, lunch, etc. Read the offering to see what's included. Normally the hostel tours don't waste time with shopping stops, as the backpacker demographic they are geared to have short patience with that sort of thing--and little discretionary money to waste.