As for recording I would just use a smart phone. I also use a micro recorder (circa 2008). Not sure the quality is as good as other methods but "good enough" for self critique.
I would address your technical performance question in a little more detail. If you are trying to solo without letting the bottom drop out I would recommend devoting some time learning the art of chord melody (the real reason the guitar exists). Arranging chord melodies teaches you exactly how to do this and is basically how classical guitar music is written and arranged (and that is the natural music of the guitar). Jazz guitarists also make extensive use of the technique. A systematic approach would involve learning chord scales (how to play up and down the scales with full 3 and 4 voice chord harmony) then applying this to simple standards you already know. The process will eventually get into your muscle memory and you will find that it becomes easier to solo with a chord melody approach. Until you get there you may want to write solos for a few tunes and then add back up chords and bass to that. It's a learning process and I'm sure you would rather just improv but it takes time to get good at the technique.
There are some great books out there on the technique and the Mel Bay series incorporates chord scales by about grade 3. There is a separate book called Guitar Melody Chord Playing System you can get on Amazon. It's very simple but effective. He goes through the simple I, IV, V7 harmonization of Major and Harmonic minor scales in every key. The sample songs are somewhat childish but that's not the point. Once you get the idea in your head and hands you can start applying it to your songs. The William Levitt method also incorporates this approach.
Many guitarists do exactly what you are trying and you could learn a lot from listening to them. I am not a country player so my examples may not fall into your taste category but the point is to learn how the technique is applied. One of the greatest Jazz guitarists was Joe Pass. He played standards solo, no back up group. He played a hollow body electric like a classical, i.e. without a pick. This brings me to the other thing you can do to make your playing richer. Learn finger style if you don't know it already. Joe would play full chords, chord melody, and even harmonize solo lines. It is worth pointing out that not every note requires a bass line! Sometimes you can just let the bottom drop for a line and pick it up again later. It's a matter of doing it tastefully. Joe does that. It is quite amazing to hear him play. You feel like you hear a group yet when you look at transcriptions there isn't much there. The Bass is thrown in once in a while but effectively. This is also the case in classical guitar music. Even though 6 strings are available a majority the music is single note lines.
As an example, I have a new student who likes folk and country and I'm teaching them chord melody. We just arranged The Gambler that way and it was fairly easy as it's mostly one chord (where it counts).
Artists who are known for chord soloing that I can recommend listening to are, Joe Pass, Bucky Pizzarelli, Van Eps, all jazz players. I would highly recommend getting a chord melody book like the Mel Bay one I mentioned above, and learn finger style. To make contact with country I know that Bluegrass guitarist use the same chord melody system as jazz players do but make use of finger picking or hybrid picking to get a quick banjo style arpeggiation of the chords as they go. So if you look up some classic bluegrass player on Youtube you may see some of that approach.
