I forgot to mention it's also easier to do it with the pedal at its lowest point. Is that what you're trying? When I first bought my Speedplays I genuinely thought I couldn't clip in but once I figured out the technique it's not been a problem - been riding them for 3 years now.

SSB,
friend of mine had a very same problem and I was trying to help him...
What I've noticed is that he was trying to push his foot down using the leg muscles. I suggested he use his weight instead and after few attempts he was fine.
So, try engaging the pedal at it's 4-5 o'clock point and lift your other leg up so your total body weight is supported only by the leg that you are trying to engage in pedal... slight twist, like you are trying to extinguish a cigarette butt, and that's it

True

but you can get light duty springs, and that is what the OP needs

No it isn't. As I said, I'm the same weight (more or less) and hardly very strong, and I have no issue. It's technique.

As patently obvious this advice may have sounded, it seems to have done the trick!

I moved the crank up to the 3 o'clock position, and then pressed down with my body weight while wiggling my foot, and by the time the crank hits the 6 o'clock position, it's clicked in. The thing I need to practice is regaining stability after putting all my weight to one side (since with Look pedals, my leg muscles seemed to do the trick without having to shift my weight to one side.)

Thanks for the help everyone!

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2012 Cannondale CAAD10

It will get easier through use,it will get to where you can clip in seated...Also use a good teflon based lube on the cleats. Speedplay even makes some.

I think this thread has most things covered so I just have two things to add from personal experience.

1 - The shims you need to get the cleat flat aren't necessarily the ones listed on the Speedplay website. Worth getting a ruler and laying it across the 3 hold adapter at a few angles to double check that it will support the cleat properly.

2 - I've also found that it's easier to clip in if I tilt my foot so that my instep engages first.

I will have to agree that the shims need to be in the right place with no cuvature at all or they won't engage properly. I have been using them a while and also figured out why the pro's are always looking for the stiffest sole with these pedals. Any flex in the sole affects getting in the pedals at various positions. My nike shoes aren't too flexy but then again my foot is small.

I'm 130 pounds, I had the same problem as you until someone told me to have the pedal down and turn your heel out as you push down and as mentioned in the begining you can use a little teflon oil on the pedal.