How to deal with my own sloppy research published when I was an undergrad?

Just don't worry about it, and keep doing your current good work.

It's pretty cool that you were able to publish a paper at all as an undergraduate. Anybody who notices your old paper will primarily notice that. The sloppiness will reflect much more on your senior co-authors (who should have been expected to know better, and educate you likewise) than yourself. And if you didn't have much more experienced co-authors, so much more cool and understandable.

Do good work now, and the sloppiness of your undergraduate paper will simply fade into insignificance.

First of all, your situation is not so unusual: in fields where student-only publications are common (like mathematics and some branches of CS), a lot of academics have one or two publications from their undergraduate days. These publications are virtually guaranteed to be significantly worse than all of their other publications. If you don't believe it, consider the logically equivalent form: "It is virtually guaranteed that the time you spend in a graduate program will significantly improve the research you do and publish."

I think a lot of people are vaguely embarrassed by their undergraduate publications: e.g. in mathematics most undergrad-only publications are in unusually undergrad-accessible fields like graph theory. But then a majority of pure mathematicians go on to study and work in much fancier fields and even (unfortunately, to be sure) to look down upon these areas. But although undergraduate publications can be embarrassing compared to later publications, they are impressive when compared to other undergraduates: a student-only undergraduate publication puts you at the head of the class.

For your paper in particular you sound more embarrassed by some slipshod details than by the work as a whole. It also sounds like you received some subpar refereeing and editorial work: in particular, asking for extra content and then not looking at the extra content is lazy. It can be surprising sometimes what apparently (and even actually, most of the time) reputable journals can publish. (You might think that each paper published in a serious journal would at least have a flawlessly grammatical title. You'd be wrong. But I digress...) What's done is done, but there's a lesson here: assuming that whatever someone else lets you publish is going to make you proud is sometimes an oversimplification and sometimes an outright error.

I would say that for the most part you should just work on feeling good rather than bad about this. The one thing that may be worth actual action on your part is making clear to interested parties that this was an undergraduate publication. There are some obvious clues to this for those who look. In my field (mathematics), up until very recently a majority of graduate students did not publish papers, so it's rather common to see gaps of approximately five years between a mathematician's first publication and all later publications. Even a gap of a few years followed by a steady stream of publication creates that impression. You should certainly have a webpage, that webpage should certainly have a CV, and that CV should certainly clearly demarcate your undergraduate years. If you feel strongly enough about it, you could list that paper separately on your webpage / CV / publication list in a category called "Undergraduate Papers".

Afterthought: if you think the flaws in your paper will cause trouble to researchers in the field, perhaps fixing the paper in some way might be in order. You could post an edited version of the paper on your webpage, and if someone cites the paper you could refer them there. But ask a more experienced academic before you do this to get a sense of whether it's really necessary.