I'm a lefty and have no problem using a rotary cutter.
If your ruler is moving when you cut, they make little sand paper disks to put under your ruler so it doesn't shift.
Maybe your table is too high and you can't get the leverage you need. Try cutting on a lower table. Are you using a 24 inch ruler? Cut only 5/6 inches at a time, move your hand up the ruler where you will need to cut, cut another 6 inch section, moving your hand up to the area you are cutting helps you keep pressure on the ruler where you are cutting, this helps keep the ruler from shifting.

If I am reading this right you are only using a regular ruler? If you are that might be making it a lot harder. I almost away use a large cutting mat that has the slots in between it.You just line up the fabric then put the rotary cutter in the slot and cut. When you are using a small mat or a ruler it's harder to get a good grip with your other hand to hold the fabric in place. (Make sure your hand isn't too close to the blade of course). I am left handed also and I do think it makes it a little harder. I am especially hard on the blades because I basically have little coordintation.

Do you happen to have 2 blades on your cutter? Sometimes those are stuck together pretty well when taking them out of the package and it looks like only one when there are actually more. That would throw you off. Just a thought....

I am left handed, have the Gingher left handed roatary cutter, have the gypsy gripper on my ruller and still cut wonky.
and no I can not cut right handed at all.
It is so frustrating.
Best of luck to you figuring this out.

I am left handed. I have some problems with cutting. I've gotten better since I watched a tutoial and noticed I wasn't resting my little finger on the mat for stability.

The one holding the ruler.

I didnt read the entire post but just wanted to say that unless I am directly over the top of what Im rotary cutting, the whole thing gets "slide-y" on me and I cant cut straight. (I cant cut from an angle). AND I apply a good amount of pressure.

Have you tried switching hands, I am left handed but I cut with my right hand. Kim

I had similar problems and finally invested in several of the June Taylor slotted rulers. I have less slippage of the ruler and better conrol of the rotary cutter because it fits in a slot so it has a sort of a track to roll in. There is also more ruler area to lay your other hand flat for holding the ruler. I have lost far less fabrice due to slippage since buying them. The largest one (23" wide) that cuts single-folded fabric is $69.99 on June Taylor's website is $65.99 on JoAnns -- but with a 50% off coupon is less than $33.00 and is shipped in a big flat cardboard container which is great for storage. This one purchase has made cutting strips so much easier and my strips are right on. These are great tools for those of us that are rotary cutter challenged. I am very short and I cannot reach across the table when cutting without moving my supporting hand and that is when the ruler moves on me or I was getting "V's" in the middle of my strips making them virtually useless, unless I was cutting them into squares.

The June Tailor strip cutter is what you need to use. It's quck, easy to learn how to do, and accurate.