OP,
read your problem and skimmed the responses.
Early in my pursuit of high end audio I had a few mis-steps. Assembling a great system takes patience and effort… especially on a budget. If this is something you plan on pursuing (with effort it can be really rewarding) I recommend buying Robert Harley’s book, The Complete Guide to High End Audio. It lays out the whole landscape. One of the difficulties here is you get advice from all sorts of perspectives from folks with vastly different experience and objectives. The book offers approach’s to assemble a very high end system for little money as well as a plan for future better systems.
My thought about your dilemma. I would assume the new integrated is a good sounding piece of equipment and therefore not the problem. Definitely let it break in. Breaking in reduces the treble / forwadness and solidifies bass. The most likely source of the problem is your source. I saw recommendations for a DAC. Good idea… I would look at Schiit Gungnir (or thinking longer termed a Yggdrasil)… probably in you price range. A good streamer makes a difference as well. But this my be longer term… think Bluesound or better.
If the DAC doesn’t do it… then save up for speakers. But the DAC is likely to give you more detail and more natural sound… so this is likely to make you very happy now.
Consider putting some photos of your current system and space under systems under your user ID. I have mine there. This helps folks help you. I mean your system could be in front of picture windows… if we saw that, the whole direction of the conversations would changed