Mazal tov on your recent wedding!
Especially at the start of marriage, the moch dachuk can be uncomfortable. However, as you note, though the moch dachuk can be omitted, it ideally should not be. When you feel ready to try again, here are some suggestions that may help:
First and foremost, try not to keep it in longer than necessary. Try to insert it right before shekiah and remove it right after tzeit hakochavim. If you will be traveling and would need to keep it in for significantly longer than is comfortable or necessary, you may omit the moch dachuk.
It might be that moistening the moch dachuk before insertion with a drop of water (so that it is damp, though not completely wet) would make it more comfortable for you. Using a vaginal moisturizer earlier in the day might also help increase comfort by providing lubrication. Sitting down for the duration of its insertion may also reduce discomfort.
It is acceptable to use a single bedikah cloth as the moch dachuk. Some women actually find it more comfortable to use a tampon wrapped with a bedikah cloth as a moch dachuk.
Please get back to us if these tips do not help.