WiFi accessories do dedicate the WiFi interface to the accessory. On phones with cellular connections, however, the phone will defer to the cellular network when connected to an access point that is not proving internet access.
Funny you ask which adapter I'd recommend. I don't endorse any specific hardware, but I did just post a rundown of the accessories I use to develop and test tesLAX, as well as some pros and cons:
Supported Hardware • tesLAX
The MX+ gives moderate performance on a single bus. One unique advantage is that it provides better background execution than other accessories, because "MFi" accessories are blessed with an additional background mode.
While I've just started testing with them, I'm impressed by the vLinker accessories. The WiFi version gives impressive data rates, and uses what I suspect may be a clone of the chip in the OBDLink MX+. (The BLE version is not yet supported by tesLAX, as BLE is not yet supported at all.)
If you're a DIY type, check out Josh Wardell's CANServer. Impressive data rates over Wifi using the Panda protocol, --- and lots of additional functionality, like on-board logging to an SD card. Works great for real-time visualization with tesLAX. It also has an ecosystem of "micro displays" and other cool add-ons.