Thought some before posting at the risk of aiding and abetting confusion on this topic.
Sure hope it doesn't. But if so, Mods feel free to get the ole axe out.
Budget sometimes has other meanings and this is one of them. Part of the budget is cost of the equipment to get the position data. The other budget is what degree of accuracy is needed and the effort needed to get that accuracy. Turning on a GPS receiver and collecting data is one thing. Having confidence in the accuracy of the data and its limitations has a little more to it.
In this case something to consider in collecting GPS positions as the boundary line is walked, is the accuracy of the GPS data. Simply put a consumer grade receiver might produce positions accurate to 10-15 meters of true position horizontal, depending on many things (signal strength, GPS satellites in view, signal deflection).
This position accuracy can be improved several ways: by collecting data when gps satellites are in best sky position for your area; augmenting the receiver GPS signals with differential correction signals from a beacon in real time; and/or correcting your receiver positions after collection by post processing the position data with software (last is not trivial and not cheap).
Realtime differential corrected positions can improve position accuracy to 3 meters (cm in more expensive equipment). The two 'free' sources of real time differential correction signals are the Coast Guard (DGPS) and the FAA (WAAS). Note there are other private DGPS services that a paid subscription based (John Deere AG Starfire is one example mentioned). Private corrections are not cheap.
Corrected vs uncorrected receiver positions can be verified by locating an
NGS benchmark and standing on it with a receiver. Comparing the NGS position with that of the receiver's uncorrected and corrected positions will give a good, quick sense of the receiver's accuracy at that time with that set of GPS satellites in view.
Coast Guard DGPS are radio beacon signals and usually good to correct positions down to 3 meters. DGPS signal reception is handled by a separate receiver (typical) or by builtin receiver in the more expensive GPS units. In any case, the GPS unit will automatically monitor and apply corrections to your positions in real time. Signal coverage for the US is shown on Coast Guard online maps and has improved a lot since the 90's. Line of sight to the Coast Guard beacons is not necessary to receive the signals and are less affected by terrain or obstructions. However, as Madhatte said, GPS satellite position signals can be compromised by tree canopy. Conifers are the biggest PITA. Hardwoods less so when bare of leaves.
WAAS, on the otherhand, is satellite based correction signals and can provide position accuracy to under 3 meters. These signals are handled by nearly all the GPS receivers directly without additional equipment. The WAAS signals are line of sight to the satellites low on the horizion and may easily be disrupted by terrain, trees, etc. These signals are more appropriate for air navigation or open ground navigation. However, it may work for your area and application. Just be aware of possible limitations.
Given your budget it might be worth looking for GPS receivers that are out of the box ready to receive WAAS and Coast Guard DGPS correction data. At least the capability is there if you need the enhanced position accuracy.
Again, if walking a boundary line once with good GPS signals and a relatively bare canopy and then plot the positions on a topo map with good confidence they are close to true positions, differential corrected positions are essential, ime.
Might have some folks in the state extension service or USDA FSA office that could be of value in helping with practical use of equipment in your area for your application. Maybe someone local that maps snowmobile trails would be worth chasing down. They might have some practical woods experience with GPS receivers and mapping within your budget that could save a lot time and money. Maybe even offer a chance to see a unit in action under a forest canopy.
Getting accurate position data reliably, as simply as possible, in one walk though is 99% of the battle. Exporting the position data and plotting it on a digital topo is icing on the corks. Just a thought.
Hope this makes some sense.
btw, 20 years ago it was a lot harder, and insanely expensive, believe me.