LSAs/LSPs (OSPF vs IS-IS speak) are flooded through the network. You are indeed correct that a LSP (I prefer ISIS) only contains information for adjacent nodes + certain IP reachability info from the node that sourced the LSP but flooding solves this and allows you to gain a full picture of the network.
You can look at the LSP DB in a router. Here's some output from a virtual environment I'm working with:
tsroot@R2-4> show isis database
IS-IS level 1 link-state database:
LSP ID Sequence Checksum Lifetime Attributes
R1-1.00-00 0x1e 0x6880 52429 L1
R1-2.00-00 0x122 0x17c4 62130 L1
R1-3.00-00 0x320 0xca73 542 L1
R1-4.00-00 0x3b8 0x9e67 761 L1
R2-1.00-00 0x26a 0x90cd 62131 L1 L2
R2-2.00-00 0x398 0x9ea4 62130 L1 L2
R2-4.00-00 0x3ce 0x8ecb 793 L1 L2 Attached
R2-5.00-00 0x2b4 0xa3b0 62137 L1 L2
R2-6.00-00 0x26b 0x78d3 62028 L1 L2
R1-5.00-00 0x1a4 0x285f 62131 L1
R1-6.00-00 0x1cc 0x6069 61984 L1
11 LSPs
We can see that we have received LSPs from all other routers in this topology and obviously we also have our local information (R2-4).
It's possible to look in more detail of the LSP:
tsroot@R2-4> show isis database R1-1 extensive
IS-IS level 1 link-state database:
R1-1.00-00 Sequence: 0x1e, Checksum: 0x6880, Lifetime: 52351 secs
IPV6 Unicast IS neighbor: R2-1.00 Metric: 30
Two-way fragment: R2-1.00-00, Two-way first fragment: R2-1.00-00
IPV6 Unicast IS neighbor: R2-2.00 Metric: 30
Two-way fragment: R2-2.00-00, Two-way first fragment: R2-2.00-00
V6 IPV6 Unicast prefix: 2001:db8::13/128 Metric: 0 Internal Up
V6 IPV6 Unicast prefix: 2001:db8:1:13:11::/112 Metric: 30 Internal Up
V6 IPV6 Unicast prefix: 2001:db8:1:13:12::/112 Metric: 30 Internal Up
Header: LSP ID: R1-1.00-00, Length: 156 bytes
Allocated length: 284 bytes, Router ID: 0.0.0.0
Remaining lifetime: 52351 secs, Level: 1, Interface: 330
Estimated free bytes: 257, Actual free bytes: 128
Aging timer expires in: 52351 secs
Protocols: IPv6
Packet: LSP ID: R1-1.00-00, Length: 156 bytes, Lifetime : 65531 secs
Checksum: 0x6880, Sequence: 0x1e, Attributes: 0x1 <L1>
NLPID: 0x83, Fixed length: 27 bytes, Version: 1, Sysid length: 0 bytes
Packet type: 18, Packet version: 1, Max area: 0
TLVs:
Area address: 49.0276 (3)
Speaks: IPV6
Topology: ipv6 unicast
Hostname: R1-1
IPv6 address: 2001:db8::13
IPV6 UnicastIS neighbor: R2-2.00, Metric: default 30
IPV6 UnicastIS neighbor: R2-1.00, Metric: default 30
IPV6 UnicastIPv6 prefix: 2001:db8::13/128 Metric 0 Up
IPV6 UnicastIPv6 prefix: 2001:db8:1:13:11::/112 Metric 30 Up
IPV6 UnicastIPv6 prefix: 2001:db8:1:13:12::/112 Metric 30 Up
No queued transmissions
IS-IS level 2 link-state database:
tsroot@R2-4>
And we can see that includes adjacency information with metric to reach those neighbours:
IPV6 Unicast IS neighbor: R2-1.00 Metric: 30
IPV6 Unicast IS neighbor: R2-2.00 Metric: 30
There's info about the directly connected IP prefixes:
V6 IPV6 Unicast prefix: 2001:db8::13/128 Metric: 0 Internal Up
V6 IPV6 Unicast prefix: 2001:db8:1:13:11::/112 Metric: 30 Internal Up
V6 IPV6 Unicast prefix: 2001:db8:1:13:12::/112 Metric: 30 Internal Up
From this information we can then compute the shortest path by starting the tree at our own node (R2-4). It's also possible to deduce loop free backup paths by rooting the tree at an adjacent node. This is called LFA (Loop Free Alternatives) and is a way of achieving IP FRR (Fast Re-Route).
The flooding of LSPs happens verbatim. Ie the LSP is not even interpreted before being flooded. It is therefore important that the size of an LSP is equal or smaller in size to the link in the network with the lowest MTU.