It might be that the TCC valve in your valve body (inside the trans) has widened the bore through use (the VB is made of aluminum, as are several other parts, but the TCC valve tends to wear the bore in particular, as it's constantly engaging / disengaging, in stock form.
If that's the case, Sonnax makes a kit that has a reamer to allow an oversize valve to be fitted. Or, you can get a new VB online (but make sure you disable TCC / PWM, either through a tune, or via a kit (a lot of people here have put in the TransGo corrective shift kit (orange box), with excellent results). But if the bore is worn in the VB, the TransGo kit won't help, as the bore has to be within operating spec for their fix to work. And the Sonnax reamer kit is expensive, and can be tricky for the lay person to use. Thankfully, some vendors sell replacement VB with the Sonnax kit already installed for you (and some rebuilders already have the reamer; it's designed for jobbers to use it over and over.)
TransGo may make an oversized TCC valve (can't remember as I write this), but if your bore is that worn, I'd go with the Sonnax - it's considered a 'permanent' fix. Again, if your bore isn't (or very slightly) worn, the TransGo kit is the easier / cheaper way to go - I have it in mine, and as I mentioned above - many here have had it in their trucks (and cars) for many years w/o issue.
Depending on how mechanically inclined you are, you can replace or upgrade the VB with the trans in-vehicle (or have a knowledgeable friend put it in for you). Plan on a weekend to have the vehicle down, taking your time and being careful. You'll need a few additional parts (like new separator plate gaskets, at a minimum.)
Yes, the TC is designed to slip, both for gear-to-gear shifts (making them 'butter soft', as manufacturers like to do), and for lockup (which can happen in 3rd, not just 4th.) The way in which this is done is called 'PWM', or Pulse Width Modulation. If you eventually get a code on this, it'll likely be P1870; you may have others, but that's the one that is most relative to this issue.
Generally, a solenoid won't fix this, and there's only one main fuse that controls the trans behavior - if it blows, or you have a break/short in the wiring harness, you'll see a bunch of codes, and the trans will likely start in 3rd gear (as 3rd is the only gear in the 4L60e that doesn't require the 'e'lectronics to shift). That's 'limp home' mode.
If the TCC bore isn't worn, and a TransGo kit doesn't fix it, then your torque converter and/or your 2-4 band may have gone south (when the trans is slipping the converter, it does this by engaging the 2-4 band on/off to 'grab' the converter.) Both of those require the trans to be dropped; depending on how much longer you plan on keeping the vehicle, you may decide to look for a less expensive option (like a used one from a junkyard, or a rebuilt one from your local Craigslist).
If you decide to do a full rebuild (or put a rebuilt one in), get a new torque converter (don't reuse the old one, and don't buy a 'reman' TC). If you go the rebuild route, you may as well have the weak areas of the 4L60e addressed (HD 2-4 band, HD clutches / steels - especially the 3-4 pack, and the sunshell). Those, along with replacing any plastic accumulators with metal (or pinless - Sonnax makes them), and a HD separator plate should give you a trans that outlasts the engine (if you got 191K out of the stocker, you probably haven't abused yours, changed the fluid / filter at least once or twice, probably don't tow with any regularity, etc.) Your builder should advise you if the pump assy. should be replaced (might be worth doing anyway, since you have almost 200K on the stocker).
Oh, and one more thing - putting in an external trans cooler is one of the best things you can do for your trans. They're not terribly expensive, and will keep your fluid under 200F - above that temperature, the life of the trans is shortened (by half, for every 10 (?) degrees above 200F.) You can put one in fairly easily, or if you're having the thing done professionally, you can have the builder plumb one in for you (a good builder will likely recommend you put one in, regardless.)
Finally, for a rebuild, putting in a good synthetic Dexron VI fluid will be cheap insurance. Valvoline makes a good one that isn't too spendy (about $8.xx a quart, retail). I use it in mine. You'll need 12qts, perhaps a 13th if you put in an external cooler that's on the larger side - I have a small 10-row stacked plate Derale that bypasses the stock cooler in the radiator entirely, and it keeps the trans at 150-160 in the summer, by itself. While you don't need a super large one (unless you're regularly towing 'heavy' or racing the thing), generally, you can never 'overcool' a transmission.