First off: kudos to you for actually providing some real information in your post (Browser and OS info). It is refreshing to not have to start out with playing 20 questions 
I use older Firefox versions and do not see that issue, and have seen no other reports of it.
>>I load ebay, but when it finishes loaded it reloads the page
Not sure what that means. Load eBay? A search page, and do you mean it reloads after you hit the search button and the search results page loads?
Does it only occur on search pages?
What sort of number string in the search box?
Does it happen every time you hit the search button, or only the first time?
Does it happen if you copy the URL from a search from the address bar, paste in a new tab, and load that?
Does it happen if you copy the URL and replace /sch/ with /dsc/ in the url and load that?
Can you post an example search URL?
All I can suggest at this point is to try another browser (Firefox, or whatever else is avail for Linux) and/or try an older version of Chrome. My initial suspicion is that it is a browser specific issue probably related to changes in the newest browser for that specific OS. Narrowing it down via experimentation is the first step to understanding, and maybe workarounds (if there are any).
If it is a specific browser issue, it ain't gonna get fixed unless by accident. In the scenario I postulate, google changed something in Chrome 68.0.3440.106 that interacts badly with eBay's code (the new changes to search?), it affects relatively few users (Linux users a small minority in the grand scheme), and is going to be item 4834 on eBay's todo list for a fix if at all.
I use older Firefox versions and do not see that issue, and have seen no other reports of it.
>>I load ebay, but when it finishes loaded it reloads the page
Not sure what that means. Load eBay? A search page, and do you mean it reloads after you hit the search button and the search results page loads?
Does it only occur on search pages?
What sort of number string in the search box?
Does it happen every time you hit the search button, or only the first time?
Does it happen if you copy the URL from a search from the address bar, paste in a new tab, and load that?
Does it happen if you copy the URL and replace /sch/ with /dsc/ in the url and load that?
Can you post an example search URL?
All I can suggest at this point is to try another browser (Firefox, or whatever else is avail for Linux) and/or try an older version of Chrome. My initial suspicion is that it is a browser specific issue probably related to changes in the newest browser for that specific OS. Narrowing it down via experimentation is the first step to understanding, and maybe workarounds (if there are any).
If it is a specific browser issue, it ain't gonna get fixed unless by accident. In the scenario I postulate, google changed something in Chrome 68.0.3440.106 that interacts badly with eBay's code (the new changes to search?), it affects relatively few users (Linux users a small minority in the grand scheme), and is going to be item 4834 on eBay's todo list for a fix if at all.
Interesting. The numbers don't ring any bells, but I suspect have some meaning.
addendum: looking at response headers for eBay page requests while composing this post, the number you posted looks very much like part of an eBay rlogid - not sure exactly what the rlogid is but *appears* to possibly be an eBay API authentication token of something similar. (Didn't find much, and not going to bother looking deeper since it really isn't important as I think it's just a symptom of another problem). Why search would be injecting that in as the search keywords is unknown.
I don't use Chrome (and won't - tried it way back when it first appeared, hated it, and never lo
looked back), so can't help with specifics.
From what I can see, the developer tools in Chrome are similar to those in Firefox, and the Network monitor in Chrome might do the trick for showing the sequence of web requests made.
I use an old addon in Firefox called HTTPFox (starting about back 10 years ago before there were developer tools built into the browser), and it readily shows that info.
(The newer Firefox and likely Chrome built in tools are more powerful, provide more info, do profiling, have filters,etc but I generally only need to see the load sequence and URLS - often to use for adding to Windows HOSTS file, IP blocklists, Adblock rules, etc - and am used to using the klunky old HTTPFox addon.)
Here is an HTTPFox capture of an example load of an store page that redirects for:
http://stores.ebay.com/SaveCentral-Outlet which redirects to https://www.ebay.com/str/savecentraloutlet
(I normally run it as a browser sidebar in the bottom 1/5 of the browser window, but detached it for screen capture purposes to show the entire sequence)
A dump of the above's upper pane with lines wrapped to not force post to be too wide (via HTTPFox's "copy all rows" context menu option)
00:00:03.341 1.598 2248 228 GET 301 Redirect to: https://www.ebay.com/str/savecentraloutlet
http://stores.ebay.com/SaveCentral-Outlet 00:00:04.948 0.616 2267 297 GET 200 text/html
https://www.ebay.com/str/savecentraloutlet 00:00:05.620 0.046 386 (7934) GET (Cache) text/css
https://ir.ebaystatic.com/rs/c/inception-6129da.css 00:00:05.623 0.100 396 (50524) GET (Cache) text/css
https://ir.ebaystatic.com/rs/c/browse-page-desktop-c34b1e.css 00:00:05.627 0.120 407 (10448) GET (Cache) text/css
https://ir.ebaystatic.com/rs/v/vbfgz414rqz5lnmzcfypj25lbyd.css?proc=DU:N 00:00:05.962 0.221 488 (16370) GET (Cache) application/x-font-woff
https://ir.ebaystatic.com/cr/v/c1/skin/v2.5.5/fonts/vq-icon-font.woff 00:00:05.970 0.376 489 (27304) GET (Cache) application/x-font-woff
https://ir.ebaystatic.com/cr/v/c1/market-sans/v1.0/MarketSans-Regular-WebS.woff 00:00:06.045 0.352 370 (41375) GET (Cache) application/x-javascript
https://ir.ebaystatic.com/rs/c/inception-7d2624.js 00:00:06.482 0.192 380 (111397) GET (Cache) application/x-javascript
https://ir.ebaystatic.com/rs/c/browse-page-desktop-168593.js 00:00:06.674 0.043 381 (34033) GET (Cache) application/x-javascript
https://ir.ebaystatic.com/rs/v/qcj0qozmfm0cdcdzkrx0lvuhsyg.js 00:00:06.729 0.025 400 (1917) GET (Cache) application/x-javascript
https://ir.ebaystatic.com/rs/c/makeebayfasterscript-src-scripts-body-78a2168a.js 00:00:06.766 0.027 381 (3255) GET (Cache) application/x-javascript
https://ir.ebaystatic.com/rs/v/10341xh50yz21mhhydueu4m5wad.js 00:00:06.793 0.027 381 (7695) GET (Cache) application/x-javascript
https://ir.ebaystatic.com/rs/v/it02syay0qyozhdaszhv1jl4yyd.js
It shows the redirection that occurs and why (a 301 redirect on http://stores.ebay.com/SaveCentral-Outlet page that points to https://www.ebay.com/str/savecentraloutlet )
Something similar might be occurring in your case.
That is the data that would be of interest if a similar procedure can be performed with Chrome's Web Developer Tools Network Monitor https://stackoverflow.com/questions/41200450/multiple-urls-copy-in-sources-network-tab
Firefox versions newer than FF31 that I run sort of have capability to copy parts of the equivalent of the HTTPFox top pane from the Network Monitor display to the clipboard, but not the same info, and from what I can see, Chrome is similar.
If the results need to be saved or posted, it looks like the only way to get all the URLS in Chrome is (was?) to run the script posted in the stackoverflow thread, or copy all as cURL which grabs a bit more than needed but gets the job done, or to take it all, including content, via the "Copy As HAR (HTTP Archive)" option, which is overkill, but appears the only way to bulk capture all the data
Can also just copy/paste just individual lines of interest one by one.
