This may be one of those opportunities where you can learn from the professionals. You might not want to use the gear they use, but many of the same concepts still apply.
1) Get the largest diameter lens and imaging chip (or 3-chip system) you can afford. The large diameter lens will allow more light into the camcorder to be processed by the imaging sensor/system. The larger imaging sensor/system will provide a much wider window of opportunity of lighting options. You mentioned "ease of use". Auto everything will be your friend - especially under ideal, bright sunlight (better yet, a light overcast day with the sunlight a little filtered).
The above would be considered "ideal lighting conditions". Pretty much any camcorder should behave well and capture really good quality video. We don't always get to work under ideal conditions, so you need to plan for that and have a wider window of opportunity.
Suggested manufacturers: Panasonic, Canon, Sony.
Be sure to record at the highest available resolution/video quality setting. You can always reduce resolution, but if you don't capture at highest resolution, you can't ever "get it back".
2) You knew enough to locate the mic element as close to the audio source (the mouth of the person speaking) as possible. In the case of a lavaliere, we all can hope that the person speaking does not turn their head/mouth away from the mic, otherwise we can have dropouts. Again, with the professionals, when wireless gear is used, the frequency range is between 470 MHz and 608 MHz - or in the 1.9 GHz range.
Wireless is challenging to do well. Other devices - and technology - are constantly transmitting so the receiver needs to be smart enough to reject spurious emissions that might be picked up as interference (static, drop-outs, etc.). Wifi and others share the 2.4 GHz spectrum... Same with the 5GHz range. And 900 MHz has been used by garage door openers, personal weather stations and other devices for a while. A few years ago (in the US), wireless mics and a few other items were kicked off the UHF and VHF TV bands being reallocated to "wireless services" (I think "Internet of Things" IoT gear). Sorry - I digress - you don't need to know about this... but it is interesting to me. There are a few other allowed wireless mic frequencies, but the takeaway is that good wireless gear is not inexpensive.
Suggested manufacturers: Sennheiser, Sony (professional), Shure, Audio Technica... there are a few others, but they can get fairly pricey. Things to look for: Change channels within a frequency range so if there is interference, moving to another channel typically resolves that. Metal case. Battery level indicator that flashes when battery power is low.
Another option is to use a field audio recorder and record the audio separately. A wired lavaliere plugged into the field recorder can work - import the audio from the recorder when editing. The video editor should have a graphic representation of the audio. Line the spikes up - when the echo is gone, the audio is synched. Use of a "slate" clap board can provide take info and the snap will provide a distinct audio spike for alignment... then mute the audio that was captured with the video.
Suggested manufacturers: Zoom, Marantz, Sony (professional), Tascam, a few others.
Another option is to use a wired mic. This can be lavaliere, handheld or shotgun (very directional). These can take some getting used to and use of XLR connectors is desirable - especially for long cable runs.
Suggested manufacturers: Audio Technica, Shure, Sennheiser.
Most camcorders less than ~$1,500 that have an audio input don't use XLR connectors. They have a single 3.5mm stereo audio input. To use XLR mics with these camcorders, an "XLR adapter" is needed. Most camcorders in this same price range will have their manual audio gain control buried in a menu - and adding an XLR adapter should include audio gain knobs and a couple of other useful features.
Suggested manufacturer: BeachTek. Since we don't know which mic(s) you might want to use, be sure to get an XLR adapter that can provide "phantom power" to the mic. This is especially important if you decide to use a shotgun mic - Decent condenser mics need power. Sometimes that means putting a AA, AAA or 9 volt battery inside the mic. Other times it means supplying the battery with power externally - phantom power...
Based on what you've told us, you seem to want to replace the camcorder. If you can keep using it under good lighting conditions, there may not be a need to do that - but the settings need to be checked - and perhaps set accordingly. If the issues continue, then replacement might be in order. I've found that camcorders in the same price range from the suggested manufacturers will provide similar video quality under good lighting conditions.
Either of the Sonys you list are OK. Panasonic and Canon will have competitive equivalents.
The mic could probably be upgraded/replaced. (Built-in mics are generally good - but they can't be in the best place for the mic to be to provide acceptable video - so the built-in mic "is not good"). I use Sennheiser wireless labs, Audio Technica and Shure XLR wired mics (sometimes with the Sennheiser wireless add-on module. My current camcorder has XLR audio inputs, so its a little easier than adding an XLR adapter.
And... if you are not using some sort of steadying device (tripods are common), then start using a steadying device. Davis & Sanford makes decent affordable tripods... like the Pro-Vista. Pretty much anything with a "fluid head" (not "fluid head like").
and taking a breath...