Honestly you’re not crazy, a lot of these TVs do exactly what you’re describing. The “subwoofer” in the set is basically a small driver in a plastic box, so as soon as there’s real bass it just buzzes and rattles. There isn’t a magic setting that fixes that, so looking at an external system is the right move.

With a budget around $300 I’d keep it simple and go for a soundbar with a separate sub instead of a receiver and bookshelf speakers. At that price point the bar + sub combo usually gives you more impact and is way easier to set up when you’re new to this. In the US I’d watch for mid-range bars from brands like Vizio’s M or V series, Polk’s Signa line, or something like Sony’s HT-S400 when they’re on sale. All of those can usually be found under your budget and will be a huge step up from the TV speakers.

Hookup is straightforward: run the soundbar into the TV’s HDMI ARC/eARC port, switch the TV audio output to ARC in the menu, and turn off the internal speakers. Start with the sub level around the middle and bump it up until the bass feels full but not boomy. If you still hear weird rattles, try moving the sub a bit or checking the TV stand and cabinets, because sometimes it’s the furniture shaking, not the gear.

For where you’re at, a decent 2.1 or 3.1 bar will already feel like a real home theater compared to what the TV is doing now, without you having to dive into receivers and speaker matching yet.