Hard to know the quality of a no-name rim, spoke combination. On a "budget" level bike, it's a fair bet that the wheelset is sub-par, in terms of basic survivability.
Two options I would consider: either have a good wheelbuilding shop do a rebuild on the thing, truing everything up; or, have a new wheelset built, using a good double-walled rim, probably a rim quite a bit wider overall than you've got now, with a decent choice of spokes/nipples. Done by a competent shop, a new wheel build can get you a bomb-proof setup.
In my own case, I had a factory Trek DS. Some years back, being a clydesdale and occasionally carrying a good 30lbs on the bike, the total would exceed 300lbs.
Opted for a wheelset with the following components: Velocity Dyad rims 700c rims, DT Swiss 2.0 spokes, DT Swiss brass nipples, with Shimano Deore M756 hubs. Very well built, strong and very true. Nary a creak or noise in several years of using them on rougher urban roads and occasional shorter dirt trails. (This setup was found on eBay for under $225 including shipping. And there are occasional deals like that these days, as well.)
You can certainly first try the truing route. It might result in a set that does fine for you. If you have issues, you can get a wheelset built to your specs.
If preferring to order a relatively inexpensive set, here's one option for a 700c build with Shimano hubs, Velocity NoBS rims, with DT Swiss spokes and DT Swiss brass nipples:
Silver Heavy Duty Velocity NoBS 700c 36h Hybrid Wheelset @ Velomine, $149. Here's the specs on the rims themselves:
Velocity NoBS 700c. I have a hard time believing that setup wouldn't be capable of working very nicely. Particularly if you select a larger tire ... say, 700x40 to 700x45 or so.