Can't get POSIX Path of given application in Applescript

Question

I've written some Applescript to create a list of paths of all applications running in the dock:

set appsRunning to []

tell application "System Events"
    repeat with p in every application process
        if background only of p is false then
            set appsRunning to appsRunning & POSIX path of (path to application p)
        end if    
    end repeat
end tell 

But when it runs I get the error - "Can't make application into type constant" and it highlights path to application p.

I don't understand why this happens because when I run

set q to POSIX path of (path to application "Finder") -- or any other application  

I get no error whatsoever and I see "/System/Library/CoreServices/Finder.app/" returned in the Results field.

How can I get this to work?

P.S. For my purposes it is essential that I get the path - the application name simply won't do. (This is because when I get the name of the application process, some of my applications which are SSBs made using Fluid return "FluidApp" as their name instead of "Gmail" or "Tumblr" or whatever site it is that I've made into an application. I need to distinguish between these and that only happens when I get the path.)

Any help would be appreciated! Thanks.

Update: I used an amended version of the first suggestion in @vadian's answer to solve my problem:

set appsRunning to {}

tell application "System Events"
    repeat with aProcess in (get application file of every application process whose background only is false)
        set appsRunning to appsRunning & POSIX path of aProcess
    end repeat
end tell

Answer 1

The element application process of System Events has a property application file, which you can get the POSIX path directly from.

set appsRunning to {}

tell application "System Events"
    repeat with aProcess in (get every application process whose background only is false)
        set end of appsRunning to POSIX path of application file of aProcess
    end repeat
end tell

or easier

tell application "System Events"
    set appsRunning to POSIX path of application file of every application process whose background only is false
end tell

additional here a solution which excludes the Finder because it runs all the time and the path is fixed

tell application "System Events"
    set appsRunning to POSIX path of application file of every application process whose background only is false and name is not "Finder"
end tell
set end of appsRunning to "/System/Library/CoreServices/Finder.app"

another solution using your original approach

set appsRunning to {}

tell application "System Events"
    set applicationNames to get name of every application process whose background only is false
end tell
repeat with aName in applicationNames
    set end of appsRunning to POSIX path of (path to application aName)
end repeat

and last but not least the AppleScriptObjC version (Mavericks and higher, in Mavericks only in a script library)

set appsRunning to (current application's NSWorkspace's sharedWorkspace()'s launchedApplications()'s valueForKey:"NSApplicationPath") as list

Though the method launchedApplications of NSWorkspace is deprecated, it works in Yosemite

to use the AppleScriptObjC in a script library save this code

use framework "Foundation"

on launchedApplications()
    return (current application's NSWorkspace's sharedWorkspace()'s launchedApplications()'s valueForKey:"NSApplicationPath") as list
end launchedApplications

as script bundle (in Mavericks you have to check "AppleScript/Objective-C library" in the side bar of the script) in ~/Library/Script Libraries. Create the folder if it doesn't exist.

Now you can call the script library from a normal script file (the script library is named "NSWorkspace.scptd")

use script "NSWorkspace"

set appsRunning to launchedApplications() of script "NSWorkspace"