I am not sure how we stop rocking her to sleep though.
In terms of the pressure you feel to solve this, reframe that pressure so that you realise you will make progress towards baby going to sleep in the cot. Because if you have expectations that someone will be able to give you a set of instructions to follow and Just Like That baby will go from awake to asleep in the cot - you are setting yourself up to fail. It doesn't work like that.
(Well, unless you just leave your baby to scream. But I'd judge you if you considered doing that to a 4 month old).
Tips and suggestions:
(1) Move all daytime naps to the bouncer.
You can then teach independent settling but still allowing for movement. After around 1h awake, park yourself on the sofa with baby in bouncer at your feet. Use your foot to rhythmically bounce (while you watch tv), reinserting dummy as needed.
Just keep going until babys mouth goes slack and dummy drops. Then slow and stop the movement. Restart movement and reinsert dummy at first sign if stirring awake, to try to lengthen the nap
You are teaching baby to sleep independently of you by doing this.
(2) Kerp the swaddle going and have a bedside cot
At night baby is going to sleep in a stationary cot. So it's harder work to get independent sleep. Accept that it's harder work, rather than getting frustrated that it's not an easy task.
Keeping swaddle use going through this phase is very helpful. It assists in calming and soothing babys thrashing limbs. But if you didn't established swaddle use consistently as a newborn, it's unsafe to introduce it now.
Bedside cot also makes life easier. Reduces you getting frustrated at having to lean over cot sides to settle baby, you can just shuffle across your bed. Again, not a magic wand to 'solve the problrm', just makes life easy as you work towards independent sleep
(3) Active dummy sucking, not passively in mouth
It's impossible to cry while simultaneously sucking. Impossible. So if you can get baby actively sucking, baby will not cry.
When giving baby dummy, every time get baby to reach for dummy actively. Do this by tickling cheek/upper lip with teat. Babys mouth will reach towards the touch. Point dummy upwards to roof of mouth when putting in. This triggers sucking.
Once in, tapping on the outside of the dummy encourages the suckibg reflex.
If after all this, baby is still crying (therefore not sucking) remove dummy and start again. Take dummy out, tickle cheek/lip and get baby to actively reach for it. Keep repeating this process until baby is actively sucking.
(4) Dummy is not meant to be used once asleep.
As baby settles, sucking movements will become less and less frequent. Don't tap dummy to keep active sucking going at this point, you'll stop baby entering deep sleep.
After a short while (5-10 mins from eyes closing) sucking will slow to stop. Mouth muscles go slack and dummy should fall out. This is baby in a deep sleep - dummy has done is job and is no longer needed until baby next wakes and needs help to ho back to sleep.
Dummy not being in babys mouth during sleep is correct and normal.
(5) Dummy resettled are normal and ok
It follows that baby will need you to put dummy on to resettle if they wake. Chances are baby will need a feed when waking, but you might be able to resettle with a dummy only.
Huge numbers of parents perceive dummy resettled as a massive problem. Thinking Tge Grass Is Greener - "if I didn't use a dummy I wouldn't be doing dummy reinsert". No - instead you'd be rocking or feeding to sleep- both far more hard work.
So accept dummy resettles will be needed for a few months. Make life easy to manage this. With a sidecar cot and multiple dummies to hand - you can master the art of dummy retrieval, reinsert, reassuring hand on babys chest to resettle - all without opening your eyes, moving from your bed or properly waking.
As babys sleep matures they can learn to link sleep cycles more effectively - so without needing and external assistance. Then you won't need to do dummy reinserts. You need to be patient and accept this is a process if learning.
Plus by about 8 ish months baby will have the dexterity to do own dummy inserts. If you were rocking/cuddling/feeding to sleep instead of using a dummy for independent sleep, by 8 months old youd be nowhere close to independent sleep. With dummy use you may well be there by that age.
Dummies are amazing. They are not instant-fix solution though. It's still a (frustrating) process you and baby have to go through.