I’d be interested in an adult Big Wheel, where you can use the little hand brake to lock and spin out.
Are you in the US?
If so maybe look here - Where To Buy a Cargo Bike in the US… Or How To DIY One! | Apartment Therapy
Canada - a quick search is turning up nothing. 
You could always make your own 
Alice learn to ride a regular bike, it really isn’t that hard. The problem with trikes is they are wiiiiide compared to a regular bike, and they are heavy. Both of these contribute to them not being a lot of fun to ride.
If you are wobbly on a regular bike, pedal faster. Honest, put on a little speed and give the bike a chance, it wants to stay up.
Oh and please always wear a helmet, regardless of what you ride.
Humm - well, I do have access to a regular bike - I guess I could practice a bit and see if I improve. Last time I tried it, it was a total disaster, but there’s no harm in trying again I guess.
speed is your friend, to a point. try to keep moving about twice walking speed. go straight and practice starting and stopping. get that good then try turns.
I’ve been riding a catrikefor years, but it isn’t like the others here.
If you feel uncomfortable on a regular bike you might want to consider a crank forward, or semi-recumbent bike.
People who feel unstable on regular bicycles often feel that way because they can’t put both feet on the ground and therefore feel they don’t have proper control. Crank forward bicycles are comfortable and allow you to put both feet on the ground. The feeling is similar to sitting on a motorcycle. I suggested one to another Doper whose useage is similar to what you’ve described: relatively short distances and trips to the grocery store. She bought one and loves it.
Most bike shops will have crank forward/semi-recumbents in stock and they’ll be happy to let you try one out to see what you think.
I’ve seen a couple of people around my small town on adult trikes, and I think those are FABULOUS! I highly endorse you getting one and “just say no” to the people who are trying to talk you into learning to ride a “real bike.” Phooey. I mean, how old do you have to be to finally get to do what YOU want? I think you will look very cool, different, maybe a bit eccentric, but also quite intriguing. And you will stand out as a woman of her own mind. Those are just style points. The practicality of the trike for short distances, hauling things, carrying groceries IMHO far surpasses a regular bike.
This brand of cargo bikes ships to Canada although they don’t appear to have three wheels: Cetma. This guy hand crafts many of his products though so maybe he would be willing to custom make a three wheeled cargo bike. The only drawback to these is they’re pretty pricey. As someone who rides a regular bike I would love to have a trike to carry stuff that’s too heavy for my bike rack/carrying on my back.
Learning how to ride a regular bike isn’t difficult at all though. When I got my bike last year (when I was 22) I hadn’t been on a bike in well over ten years. I was also very nervous because I’d never had a properly sized bike, so I wasn’t used to riding and not being able to touch the ground while on the saddle. The guy who sold me the bike had to re-teach me how to ride (I had some embarrassing falls and many bruises) and after lots of practice it’s now second nature, so I really think you can do it. There’s also nothing wrong with starting out with a bike that’s a little too small so you can reach the ground and feel safer- riding a bike with a higher saddle just makes pedaling more efficient since your leg extends all the way.
But otherwise, go for the trike- I see them parked on the street all the time so clearly many people find them useful.
You can get adult strength training wheels on a bike until you are comfortable on your own.
I feel what you feel.
As a child, I never learned to successfully master the complicated bike riding technique that the other children seemed to excel at. I watched from my window as the popped wheelies and rode around the neighborhood gleefully, and I plotted, the day, the rise of the adult tricycles.
My first adult tricycle was engineered out of bits and pieces I found lying around, spare tires, an old steering wheel, etc. Because of the time I invested into this, I became extremely awkward and shunned by the world. My mother has accepted and supported me throughout my journey.
I have engineered over thirty adult tricycles of my own already. I will give you one free of charge if you promise to spread the word. We will gain the respect we deserve some day, friend.
If you insist on a three-wheeler, I’ll echo Runner Pat and recommend one with two wheels in front. They are usually much more stable than regular trikes.
A cargo bike is the way to go if you want to transport a kid around. I ride a Christiania Bike and I love it. It holds 2 kids + shopping. There are other brands as well, specially made for transporting kids - like the one LotusPotus mentions - but I’d go for a multi-purpose one any day.
There some somewhat more affordable tadpole trikes to be had from Terra Trike, a few in the sub-$1000 range. They are not “hard core” as such, these trikes are rather popular with people who like a bit of comfort with their exercise. It takes some getting used to, riding that low, but you would still be big enough for drivers to see – if they can see a dog, they can see you. And they are really fun, being low makes you feel like you are moving fast.
Alternately, there are a few very nice “delta” trikes, like what they have over at Hase. Those start well north of 4 figures, if you can see spending that much, and tend to be slightly better for cargo. The cool thing about most recumbent deltas is that you can get hitches to take off the front wheel and link them together like a tandem with separate chains. IIRC, Hase set a record for a train with over 60 trikes linked up, but the hitches are not brand specific, different models can easily be linked.
However, for grocery shopping, security is a big concern. If you spend that kind of money on a trike, can you be confident it will be there when you come out? Will a simple lock be enough? Or could you maybe convince the store to let you bring it inside with you?
Saw someone with adult training wheeels, said they got 'em @ the Wal-place
http://www.walmart.com/ip/Bicycle-Wheel-Stabilizer-Kit/15734254
The ultimate adult trike is the Bluevelo Quest, which is made in Canada (under license from a Dutch company that designed it).
If you’re not ready to spend quite that much, TerraTrike and Catrike mentioned above are both very good companies. TerraTrike Rover is $699, and is probably the best <$1000 recumbent trike. Greenspeed and ICE are also good, but a bit more expensive.