Depends on how cheap you want to be. Use ethernet and disable all but one of the dhcp servers plugging the leaves LAN port to LAN port. Or just buy repeaters. I tend to have had bad experiences with cheap wifi routers with or without third party firmware.
DD-WRT/tomato/others do have mesh features but it depends on the radio chipset and other things.
I've tried doing this before and my devices "stick" to farther away access points as I walk across the house. Are there ways to solve this problem, and/or do the solutions mentioned in the article solve this problem?
No and probably not. The biggest issue with wifi is almost always the station/client. Clever units may send de-auth packets to try to get you to renegotiate with an AP with a stronger signal but nothing is guaranteed and some client stacks can be stubborn
The higher-end APs (like the aformentioned UniFi) do have a transparent roaming feature, where every AP broadcasts the same MAC, and the APs handle traffic handoff. No idea how well it works.
DD-WRT/tomato/others do have mesh features but it depends on the radio chipset and other things.
You can just buy a 802.11ac repeater for $40.